Complexity Número especial da Nature Physics Insight sobre complexidade

COMPLEXITY

JANUARY 2012

NATURE PHYSICS | VOL 8 | JANUARY 2012 | wwwnaturecomnaturephysics 13

INSIGHT | CONTENTS

NPG LONDONThe Macmillan Building 4 Crinan Street London N1 9XWT +44 207 833 4000 F +44 207 843 4563naturephysicsnaturecom

EDITOR ALISON WRIGHT

INSIGHT EDITOR ANDREAS TRABESINGER

PRODUCTION EDITOR JENNY MARSDEN

COPY EDITOR MELANIE HANVEY

ART EDITOR KAREN MOORE

EDITORIAL ASSISTANTAMANDA WYATT

MARKETINGSARAH-JANE BALDOCK

PUBLISHERRUTH WILSON

EDITOR-IN-CHIEF NATURE PUBLICATIONSPHILIP CAMPBELL

COVER IMAGE In many large ensembles the property of the system as a whole cannot be understood by studying the individual entities mdash neurons in the brain for example or transport users in traffic networks The past decade however has seen important progress in our fundamental understanding of what such seemingly disparate lsquocomplex systemsrsquo have in common Image copy Marvin E NewmanGetty Images

A formal definition of what constitutes a complex system is not easy to devise equally difficult is the

delineation of which fields of study fall within the bounds of lsquocomplexityrsquo An appealing approach mdash but only one of several possibilities mdash is to play on the lsquomore is differentrsquo theme declaring that the properties of a complex system as a whole cannot be understood from the study of its individual constituents There are many examples from neurons in the brain to transport users in traffic networks to data packages in the Internet

Large datasets mdash collected for example in proteomic studies or captured in records of mobile-phone users and Internet traffic mdash now provide an unprecedented level of information about these systems Indeed the availability of these detailed datasets has led to an explosion of activity in the modelling of complex systems Data-based models can not only provide an understanding of the properties and behaviours of individual systems but also beyond that might lead to the discovery of common properties between seemingly disparate systems

Much of the progress made during the past decade or so comes under the banner of lsquonetwork sciencersquo The representation of complex systems as networks or graphs

has proved to be a tremendously useful abstraction and has led to an understanding of how many real-world systems are structured what kinds of dynamic processes they support and how they interact with each other This Nature Physics Insight is therefore admittedly inclined towards research in complex networks As Albert-Laacuteszloacute Barabaacutesi argues in his Commentary the past decade has indeed witnessed a lsquonetwork takeoverrsquo On the other hand James Crutchfield in his review of the tools for discovering patterns and quantifying their structural complexity demonstrates beautifully how fundamental theories of information and computation have led to a deeper understanding of just what lsquocomplex systemsrsquo are

For a topic as broad as complexity it is impossible to do justice to all of the recent developments The field has been shaped over decades by advances in physics engineering computer science biology and sociology and its ramifications are equally diverse But a selection had to be made and we hope that this Insight will prove inspiring and a showcase for the pivotal role that physicists are playing mdash and are bound to play mdash in the inherently multidisciplinary endeavour of making sense of complexity

Andreas Trabesinger Senior Editor

Complexity

COMMENTARYThe network takeoverAlbert-Laacuteszloacute Barabaacutesi 14

REVIEW ARTICLESBetween order and chaosJames P Crutchfield 17Communities modules and large-scale structure in networks M E J Newman 25Modelling dynamical processes in complex socio-technical systemsAlessandro Vespignani 32

PROGRESS ARTICLENetworks formed from interdependent networksJianxi Gao Sergey V Buldyrev H Eugene Stanley and Shlomo Havlin 40

14 NATURE PHYSICS | VOL 8 | JANUARY 2012 | wwwnaturecomnaturephysics

COMMENTARY | INSIGHT

The network takeoverAlbert-Laacuteszloacute Barabaacutesi

Reductionism as a paradigm is expired and complexity as a field is tired Data-based mathematical models of complex systems are offering a fresh perspective rapidly developing into a new discipline network science

Reports of the death of reductionism are greatly exaggerated It is so ingrained in our thinking that if one day some

magical force should make us all forget it we would promptly have to reinvent it The real worry is not with reductionism which as a paradigm and tool is rather useful It is necessary but no longer sufficient But weighing up better ideas it became a burden

ldquoYou never want a serious crisis to go to wasterdquo Ralph Emmanuel at that time Obamarsquos chief of staff famously proclaimed in November 2008 at the height of the financial meltdown Indeed forced by an imminent need to go beyond reductionism a new network-based paradigm is emerging that is taking science by storm It relies on datasets that are inherently incomplete and noisy It builds on a set of sharp tools developed during the past decade that seem to be just as useful in search engines as in cell biology It is making a real impact from science to industry Along the way it

points to a new way to handle a century-old problem complexity

A better understanding of the pieces cannot solve the difficulties that many research fields currently face from cell biology to software design There is no lsquocancer genersquo A typical cancer patient has mutations in a few dozen of about 300 genes an elusive combinatorial problem whose complexity is increasingly a worry to the medical community No single regulation can legislate away the economic malady that is slowly eating at our wealth It is the web of diverging financial and political interests that makes policy so difficult to implement Consciousness cannot be reduced to a single neuron It is an emergent property that engages billions of synapses In fact the more we know about the workings of individual genes banks or neurons the less we understand the system as a whole Consequently an increasing number of the big questions of contemporary

science are rooted in the same problem we hit the limits of reductionism No need to mount a defence of it Instead we need to tackle the real question in front of us complexity

The complexity argument is by no means new It has re-emerged repeatedly during the past decades The fact that it is still fresh underlines the lack of progress achieved so far It also stays with us for good reason complexity research is a thorny undertaking First its goals are easily confusing to the outsider What does it aim to address mdash the origins of social order biological complexity or economic interconnectedness Second decades of research on complexity were driven by big sweeping theoretical ideas inspired by toy models and differential equations that ultimately failed to deliver Think synergetics and its slave modes think chaos theory ultimately telling us more about unpredictability than how to predict nonlinear systems think self-organized criticality a sweeping collection of scaling ideas squeezed into a sand pile think fractals hailed once as the source of all answers to the problems of pattern formation We learned a lot but achieved little our tools failed to keep up with the shifting challenges that complex systems pose Third there is a looming methodological question what should a theory of complexity deliver A new Maxwellian formula condensing into a set of elegant equations every ill that science faces today Or a new uncertainty principle encoding what we can and what we canrsquot do in complex systems Finally who owns the science of complexity Physics Engineering Biology mathematics computer science All of the above Anyone

These questions have resisted answers for decades Yet something has changed in the past few years The driving force behind this change can be condensed into a single word data Fuelled by cheap sensors and high-throughput technologies

Network universe A visualization of the first large-scale network explicitly mapped out to explore the large-scale structure of real networks The map was generated in 1999 and represents a small portion of the World Wide Web11 this map has led to the discovery of scale-free networks Nodes are web documents links correspond to URLs Visualization by Mauro Martino Alec Pawling and Chaoming Song

INSIGHT | COMMENTARY

the data explosion that we witness today from social media to cell biology is offering unparalleled opportunities to document the inner workings of many complex systems Microarray and proteomic tools offer us the simultaneous activity of all human genes and proteins mobile-phone records capture the communication and mobility patterns of whole countries1 importndashexport and stock data condense economic activity into easily accessible databases2 As scientists sift through these mountains of data we are witnessing an increasing awareness that if we are to tackle complexity the tools to do so are being born right now in front of our eyes The field that benefited most from this data windfall is often called network theory and it is fundamentally reshaping our approach to complexity

Born at the twilight of the twentieth century network theory aims to understand the origins and characteristics of networks that hold together the components in various complex systems By simultaneously looking at the World Wide Web and genetic networks Internet and social systems it led to the discovery that despite the many differences in the nature of the nodes and the interactions between them the networks behind most complex systems are governed by a series of fundamental laws that determine and limit their behaviour

On the surface network theory is prone to the failings of its predecessors It has its own big ideas from scale-free networks to the theory of network evolution3 from community formation45 to dynamics on networks6 But there is a defining difference These ideas have not been gleaned from toy models or mathematical anomalies They are based on data and meticulous observations The theory of evolving networks was motivated by extensive empirical evidence documenting the scale-free nature of the degree distribution from the cell to the World Wide Web the formalism behind degree correlations was preceded by data documenting correlations on the Internet and on cellular maps78 the extensive theoretical work on spreading processes

was preceded by decades of meticulous data collection on the spread of viruses and fads gaining a proper theoretical footing in the network context6 This data-inspired methodology is an important shift compared with earlier takes on complex systems Indeed in a survey of the ten most influential papers in complexity it will be difficult to find one that builds directly on experimental data In contrast among the ten most cited papers in network theory you will be hard pressed to find one that does not directly rely on empirical evidence

With its deep empirical basis and its host of analytical and algorithmic tools today network theory is indispensible in the study of complex systems We will never understand the workings of a cell if we ignore the intricate networks through which its proteins and metabolites interact with each other We will never foresee economic meltdowns unless we map out the web of indebtedness that characterizes the financial system These profound changes in complexity research echo major economic and social shifts The economic giants of our era are no longer carmakers and oil producers but the companies that build manage or fuel our networks Cisco Google Facebook Apple or Twitter Consequently during the past decade question by question and system by system network science has hijacked complexity research Reductionism deconstructed complex systems bringing us a theory of individual nodes and links Network theory is painstakingly reassembling them helping us to see the whole again One thing is increasingly clear no theory of the cell of social media or of the Internet can ignore the profound network effects that their interconnectedness cause Therefore if we are ever to have a theory of complexity it will sit on the shoulders of network theory

The daunting reality of complexity research is that the problems it tackles are so diverse that no single theory can satisfy all needs The expectations of social scientists for a theory of social complexity are quite different from the questions posed by biologists as they seek to uncover the phenotypic heterogeneity of cardiovascular disease We may however follow in the footsteps of Steve Jobs who once insisted that it is not the consumerrsquos job to know what they want It is our job those of us working on the mathematical theory of complex systems to define the science of the complex Although no theory can satisfy all needs what we can strive for is a broad framework within which most needs can be addressed

The twentieth century has witnessed the birth of such a sweeping enabling framework quantum mechanics Many advances of the century from electronics to astrophysics from nuclear energy to quantum computation were built on the theoretical foundations that it offered In the twenty-first century network theory is emerging as its worthy successor it is building a theoretical and algorithmic framework that is energizing many research fields and it is closely followed by many industries As network theory develops its mathematical and intellectual core it is becoming an indispensible platform for science business and security helping to discover new drug targets delivering Facebookrsquos latest algorithms and aiding the efforts to halt terrorism

As physicists we cannot avoid the elephant in the room what is the role of physics in this journey We physicists do not have an excellent track record in investing in our future For decades we forced astronomers into separate departments under the slogan it is not physics Now we bestow on them our highest awards such as last yearrsquos Nobel Prize For decades we resisted biological physics exiling our brightest colleagues to medical schools Along the way we missed out on the bio-revolution bypassing the financial windfall that the National Institutes of Health bestowed on biological complexity proudly shrinking our physics departments instead We let materials science be taken over by engineering schools just when the science had matured enough to be truly lucrative Old reflexes never die making many now wonder whether network science is truly physics The answer is obvious it is much bigger than physics Yet physics is deeply entangled with it the Institute for Scientific Information (ISI) highlighted two network papers39 among the ten most cited physics papers of the past decade and in about a year Chandrashekharrsquos 1945 tome which has been the most cited paper in Review of Modern Physics for decades will be dethroned by a decade-old paper on network theory10 Physics has as much to offer to this journey as it has to benefit from it

Although physics has owned complexity research for many decades it is not without competition any longer Computer science fuelled by its poster progenies

An increasing number of the big questions of contemporary science are rooted in the same problem we hit the limits of reductionism

Who owns the science of complexity

such as Google or Facebook is mounting a successful attack on complexity fuelled by the conviction that a sufficiently fast algorithm can tackle any problem no matter how complex This confidence has prompted the US Directorate for Computer and Information Science and Engineering to establish the first network-science programme within the US National Science Foundation Bioinformatics with its rich resources backed by the National Institutes of Health is pushing from a different direction aiming to quantify biological complexity Complexity and network science need both the intellectual and financial resources that different communities can muster But as the field enters the spotlight physics must assert its engagement if it wants to continue to be present at the table

As I follow the debate surrounding the faster-than-light neutrinos I wish deep

down for it to be true Physics needs the shot in the arm that such a development could deliver Our children no longer want to become physicists and astronauts They want to invent the next Facebook instead Short of that they are happy to land a job at Google They donrsquot talk quanta mdash they dream bits They donrsquot see entanglement but recognize with ease nodes and links As complexity takes a driving seat in science engineering and business we physicists cannot afford to sit on the sidelines We helped to create it We owned it for decades We must learn to take pride in it And this means as our forerunners did a century ago with quantum mechanics that we must invest in it and take it to its conclusion

Albert-Laacuteszloacute Barabaacutesi is at the Center for Complex Network Research and Departments of Physics Computer Science and Biology Northeastern

University Boston Massachusetts 02115 USA the Center for Cancer Systems Biology Dana-Farber Cancer Institute Boston Massachusetts 02115 USA and the Department of Medicine Brigham and Womenrsquos Hospital Harvard Medical School Boston Massachusetts 02115 USA e-mail albneuedu

References1 Onnela J P et al Proc Natl Acad Sci USA

104 7332ndash7336 (2007)2 Hidalgo C A Klinger B Barabaacutesi A L amp Hausmann R

Science 317 482ndash487 (2007)3 Barabaacutesi A L amp Albert R Science 286 509ndash512 (1999)4 Newman M E J Networks An Introduction (Oxford Univ

Press 2010)5 Palla G Farkas I J Dereacutenyi I amp Vicsek T Nature

435 814ndash818 (2005)6 Pastor-Satorras R amp Vespignani A Phys Rev Lett

86 3200ndash3203 (2001)7 Pastor-Satorras R Vaacutezquez A amp Vespignani A Phys Rev Lett

87 258701 (2001)8 Maslov S amp Sneppen K Science 296 910ndash913 (2002)9 Watts D J amp Strogatz S H Nature 393 440ndash442 (1998)10 Barabaacutesi A L amp Albert R Rev Mod Phys 74 47ndash97 (2002)11 Albert R Jeong H amp Barabaacutesi A-L Nature 401 130-131 (1999)

INSIGHT |REVIEW ARTICLESPUBLISHED ONLINE 22 DECEMBER 2011 | DOI 101038NPHYS2190

Between order and chaosJames P Crutchfield

What is a pattern How dowe come to recognize patterns never seen before Quantifying the notion of pattern and formalizingthe process of pattern discovery go right to the heart of physical science Over the past few decades physicsrsquo view of naturersquoslack of structuremdashits unpredictabilitymdashunderwent a major renovation with the discovery of deterministic chaos overthrowingtwo centuries of Laplacersquos strict determinism in classical physics Behind the veil of apparent randomness though manyprocesses are highly ordered following simple rules Tools adapted from the theories of information and computation havebrought physical science to the brink of automatically discovering hidden patterns and quantifying their structural complexity

One designs clocks to be as regular as physically possible Somuch so that they are the very instruments of determinismThe coin flip plays a similar role it expresses our ideal of

the utterly unpredictable Randomness is as necessary to physicsas determinismmdashthink of the essential role that lsquomolecular chaosrsquoplays in establishing the existence of thermodynamic states Theclock and the coin flip as such are mathematical ideals to whichreality is often unkind The extreme difficulties of engineering theperfect clock1 and implementing a source of randomness as pure asthe fair coin testify to the fact that determinism and randomness aretwo inherent aspects of all physical processes

In 1927 van der Pol a Dutch engineer listened to the tonesproduced by a neon glow lamp coupled to an oscillating electricalcircuit Lacking modern electronic test equipment he monitoredthe circuitrsquos behaviour by listening through a telephone ear pieceIn what is probably one of the earlier experiments on electronicmusic he discovered that by tuning the circuit as if it were amusical instrument fractions or subharmonics of a fundamentaltone could be produced This is markedly unlike common musicalinstrumentsmdashsuch as the flute which is known for its purity ofharmonics or multiples of a fundamental tone As van der Poland a colleague reported in Nature that year2 lsquothe turning of thecondenser in the region of the third to the sixth subharmonicstrongly reminds one of the tunes of a bag pipersquo

Presciently the experimenters noted that when tuning the circuitlsquooften an irregular noise is heard in the telephone receivers beforethe frequency jumps to the next lower valuersquoWe nowknow that vander Pol had listened to deterministic chaos the noise was producedin an entirely lawful ordered way by the circuit itself The Naturereport stands as one of its first experimental discoveries Van der Poland his colleague van der Mark apparently were unaware that thedeterministic mechanisms underlying the noises they had heardhad been rather keenly analysed three decades earlier by the Frenchmathematician Poincareacute in his efforts to establish the orderliness ofplanetary motion3ndash5 Poincareacute failed at this but went on to establishthat determinism and randomness are essential and unavoidabletwins6 Indeed this duality is succinctly expressed in the twofamiliar phrases lsquostatisticalmechanicsrsquo and lsquodeterministic chaosrsquo

Complicated yes but is it complexAs for van der Pol and van der Mark much of our appreciationof nature depends on whether our mindsmdashor more typically thesedays our computersmdashare prepared to discern its intricacies Whenconfronted by a phenomenon for which we are ill-prepared weoften simply fail to see it although we may be looking directly at it

Complexity Sciences Center and Physics Department University of California at Davis One Shields Avenue Davis California 95616 USAe-mail chaosucdavisedu

Perception is made all the more problematic when the phenomenaof interest arise in systems that spontaneously organize

Spontaneous organization as a common phenomenon remindsus of a more basic nagging puzzle If as Poincareacute found chaos isendemic to dynamics why is the world not a mass of randomnessThe world is in fact quite structured and we now know severalof the mechanisms that shape microscopic fluctuations as theyare amplified to macroscopic patterns Critical phenomena instatistical mechanics7 and pattern formation in dynamics89 aretwo arenas that explain in predictive detail how spontaneousorganization works Moreover everyday experience shows us thatnature inherently organizes it generates pattern Pattern is as muchthe fabric of life as lifersquos unpredictability

In contrast to patterns the outcome of an observation ofa random system is unexpected We are surprised at the nextmeasurement That surprise gives us information about the systemWe must keep observing the system to see how it is evolving Thisinsight about the connection between randomness and surprisewas made operational and formed the basis of the modern theoryof communication by Shannon in the 1940s (ref 10) Given asource of random events and their probabilities Shannon defined aparticular eventrsquos degree of surprise as the negative logarithm of itsprobability the eventrsquos self-information is Ii=minuslog2pi (The unitswhen using the base-2 logarithm are bits) In this way an eventsay i that is certain (pi = 1) is not surprising Ii = 0 bits Repeatedmeasurements are not informative Conversely a flip of a fair coin(pHeads= 12) is maximally informative for example IHeads= 1 bitWith each observation we learn in which of two orientations thecoin is as it lays on the table

The theory describes an information source a random variableX consisting of a set i = 0 1 k of events and theirprobabilities pi Shannon showed that the averaged uncertaintyH [X ] =

sumi piIimdashthe source entropy ratemdashis a fundamental

property that determines how compressible an informationsourcersquos outcomes are

With information defined Shannon laid out the basic principlesof communication11 He defined a communication channel thataccepts messages from an information source X and transmitsthem perhaps corrupting them to a receiver who observes thechannel output Y To monitor the accuracy of the transmissionhe introduced the mutual information I [X Y ] =H [X ]minusH [X |Y ]between the input and output variables The first term is theinformation available at the channelrsquos input The second termsubtracted is the uncertainty in the incoming message if thereceiver knows the output If the channel completely corrupts so

REVIEW ARTICLES | INSIGHT NATURE PHYSICS DOI101038NPHYS2190

that none of the source messages accurately appears at the channelrsquosoutput then knowing the output Y tells you nothing about theinput and H [X |Y ] = H [X ] In other words the variables arestatistically independent and so the mutual information vanishesIf the channel has perfect fidelity then the input and outputvariables are identical what goes in comes out The mutualinformation is the largest possible I [X Y ] = H [X ] becauseH [X |Y ] = 0 The maximum inputndashoutput mutual informationover all possible input sources characterizes the channel itself andis called the channel capacity

C =maxP(X)

I [X Y ]

Shannonrsquos most famous and enduring discovery thoughmdashonethat launched much of the information revolutionmdashis that aslong as a (potentially noisy) channelrsquos capacity C is larger thanthe information sourcersquos entropy rate H [X ] there is way toencode the incoming messages such that they can be transmittederror free11 Thus information and how it is communicated weregiven firm foundation

How does information theory apply to physical systems Letus set the stage The system to which we refer is simply theentity we seek to understand by way of making observationsThe collection of the systemrsquos temporal behaviours is the processit generates We denote a particular realization by a time seriesof measurements xminus2xminus1x0x1 The values xt taken at eachtime can be continuous or discrete The associated bi-infinitechain of random variables is similarly denoted except usinguppercase Xminus2Xminus1X0X1 At each time t the chain has a pastXt = Xtminus2Xtminus1 and a future X=XtXt+1 We will also refer toblocksXt prime=XtXt+1 Xt primeminus1tlt t prime The upper index is exclusive

To apply information theory to general stationary processes oneuses Kolmogorovrsquos extension of the source entropy rate1213 Thisis the growth rate hmicro

hmicro= lim`rarrinfin

H (`)`

where H (`)=minussumx`Pr(x`)log2Pr(x`) is the block entropymdashthe

Shannon entropy of the length-` word distribution Pr(x`) hmicrogives the sourcersquos intrinsic randomness discounting correlationsthat occur over any length scale Its units are bits per symboland it partly elucidates one aspect of complexitymdashthe randomnessgenerated by physical systems

We now think of randomness as surprise and measure its degreeusing Shannonrsquos entropy rate By the same token can we saywhat lsquopatternrsquo is This is more challenging although we knoworganization when we see it

Perhaps one of the more compelling cases of organization isthe hierarchy of distinctly structured matter that separates thesciencesmdashquarks nucleons atoms molecules materials and so onThis puzzle interested Philip Anderson who in his early essay lsquoMoreis differentrsquo14 notes that new levels of organization are built out ofthe elements at a lower level and that the new lsquoemergentrsquo propertiesare distinct They are not directly determined by the physics of thelower level They have their own lsquophysicsrsquo

This suggestion too raises questions what is a lsquolevelrsquo andhow different do two levels need to be Anderson suggested thatorganization at a given level is related to the history or the amountof effort required to produce it from the lower level As we will seethis can be made operational

ComplexitiesTo arrive at that destination we make two main assumptions Firstwe borrowheavily fromShannon every process is a communicationchannel In particular we posit that any system is a channel that

communicates its past to its future through its present Second wetake into account the context of interpretation We view buildingmodels as akin to decrypting naturersquos secrets How do we cometo understand a systemrsquos randomness and organization given onlythe available indirect measurements that an instrument providesTo answer this we borrow again from Shannon viewing modelbuilding also in terms of a channel one experimentalist attemptsto explain her results to another

The following first reviews an approach to complexity thatmodels system behaviours using exact deterministic representa-tions This leads to the deterministic complexity and we willsee how it allows us to measure degrees of randomness Afterdescribing its features and pointing out several limitations theseideas are extended to measuring the complexity of ensembles ofbehavioursmdashto what we now call statistical complexity As wewill see it measures degrees of structural organization Despitetheir different goals the deterministic and statistical complexitiesare related and we will see how they are essentially complemen-tary in physical systems

Solving Hilbertrsquos famous Entscheidungsproblem challenge toautomate testing the truth of mathematical statements Turingintroduced a mechanistic approach to an effective procedurethat could decide their validity15 The model of computationhe introduced now called the Turing machine consists of aninfinite tape that stores symbols and a finite-state controller thatsequentially reads symbols from the tape and writes symbols to itTuringrsquos machine is deterministic in the particular sense that thetape contents exactly determine the machinersquos behaviour Giventhe present state of the controller and the next symbol read off thetape the controller goes to a unique next state writing at mostone symbol to the tape The input determines the next step of themachine and in fact the tape input determines the entire sequenceof steps the Turing machine goes through

Turingrsquos surprising result was that there existed a Turingmachine that could compute any inputndashoutput functionmdashit wasuniversal The deterministic universal Turing machine (UTM) thusbecame a benchmark for computational processes

Perhaps not surprisingly this raised a new puzzle for the originsof randomness Operating from a fixed input could a UTMgenerate randomness orwould its deterministic nature always showthrough leading to outputs that were probabilistically deficientMore ambitiously could probability theory itself be framed in termsof this new constructive theory of computation In the early 1960sthese and related questions led a number of mathematiciansmdashSolomonoff1617 (an early presentation of his ideas appears inref 18) Chaitin19 Kolmogorov20 andMartin-Loumlf21mdashtodevelop thealgorithmic foundations of randomness

The central question was how to define the probability of a singleobject More formally could a UTM generate a string of symbolsthat satisfied the statistical properties of randomness The approachdeclares that models M should be expressed in the language ofUTM programs This led to the KolmogorovndashChaitin complexityKC(x) of a string x The KolmogorovndashChaitin complexity is thesize of the minimal program P that generates x running ona UTM (refs 1920)

KC(x)= argmin|P| UTM P = x

One consequence of this should sound quite familiar by nowIt means that a string is random when it cannot be compressed arandom string is its own minimal program The Turing machinesimply prints it out A string that repeats a fixed block of lettersin contrast has small KolmogorovndashChaitin complexity The Turingmachine program consists of the block and the number of times itis to be printed Its KolmogorovndashChaitin complexity is logarithmic

NATURE PHYSICS DOI101038NPHYS2190 INSIGHT | REVIEW ARTICLES

in the desired string length because there is only one variable partof P and it stores log ` digits of the repetition count `

Unfortunately there are a number of deep problems withdeploying this theory in a way that is useful to describing thecomplexity of physical systems

First KolmogorovndashChaitin complexity is not a measure ofstructure It requires exact replication of the target string ThereforeKC(x) inherits the property of being dominated by the randomnessin x Specifically many of the UTM instructions that get executedin generating x are devoted to producing the lsquorandomrsquo bits of x Theconclusion is that KolmogorovndashChaitin complexity is a measure ofrandomness not a measure of structure One solution familiar inthe physical sciences is to discount for randomness by describingthe complexity in ensembles of behaviours

Furthermore focusing on single objects was a feature not abug of KolmogorovndashChaitin complexity In the physical scienceshowever this is a prescription for confusion We often haveaccess only to a systemrsquos typical properties and even if we hadaccess to microscopic detailed observations listing the positionsand momenta of molecules is simply too huge and so useless adescription of a box of gas In most cases it is better to know thetemperature pressure and volume

The issue is more fundamental than sheer system size arisingevenwith a few degrees of freedom Concretely the unpredictabilityof deterministic chaos forces the ensemble approach on us

The solution to the KolmogorovndashChaitin complexityrsquos focus onsingle objects is to define the complexity of a systemrsquos processmdashtheensemble of its behaviours22 Consider an information sourcethat produces collections of strings of arbitrary length Givena realization x` of length ` we have its KolmogorovndashChaitincomplexity KC(x`) of course but what can we say about theKolmogorovndashChaitin complexity of the ensemble x` First defineits average in terms of samples x i

` i=1M

KC(`)=〈KC(x`)〉= limMrarrinfin

Msumi=1

KC(x i`)

How does the KolmogorovndashChaitin complexity grow as a functionof increasing string length For almost all infinite sequences pro-duced by a stationary process the growth rate of the KolmogorovndashChaitin complexity is the Shannon entropy rate23

KC(`)`

As a measuremdashthat is a number used to quantify a systempropertymdashKolmogorovndashChaitin complexity is uncomputable2425There is no algorithm that taking in the string computes itsKolmogorovndashChaitin complexity Fortunately this problem iseasily diagnosed The essential uncomputability of KolmogorovndashChaitin complexity derives directly from the theoryrsquos clever choiceof a UTM as themodel class which is so powerful that it can expressundecidable statements

One approach to making a complexity measure constructiveis to select a less capable (specifically non-universal) class ofcomputationalmodelsWe can declare the representations to be forexample the class of stochastic finite-state automata2627 The resultis a measure of randomness that is calibrated relative to this choiceThus what one gains in constructiveness one looses in generality

Beyond uncomputability there is the more vexing issue ofhow well that choice matches a physical system of interest Evenif as just described one removes uncomputability by choosinga less capable representational class one still must validate thatthese now rather specific choices are appropriate to the physicalsystem one is analysing

At themost basic level the Turingmachine uses discrete symbolsand advances in discrete time steps Are these representationalchoices appropriate to the complexity of physical systems Whatabout systems that are inherently noisy those whose variablesare continuous or are quantum mechanical Appropriate theoriesof computation have been developed for each of these cases2829although the original model goes back to Shannon30 More tothe point though do the elementary components of the chosenrepresentational scheme match those out of which the systemitself is built If not then the resulting measure of complexitywill be misleading

Is there a way to extract the appropriate representation from thesystemrsquos behaviour rather than having to impose it The answercomes not from computation and information theories as abovebut from dynamical systems theory

Dynamical systems theorymdashPoincareacutersquos qualitative dynamicsmdashemerged from the patent uselessness of offering up an explicit listof an ensemble of trajectories as a description of a chaotic systemIt led to the invention of methods to extract the systemrsquos lsquogeometryfrom a time seriesrsquo One goal was to test the strange-attractorhypothesis put forward byRuelle andTakens to explain the complexmotions of turbulent fluids31

How does one find the chaotic attractor given a measurementtime series from only a single observable Packard and othersproposed developing the reconstructed state space from successivetime derivatives of the signal32 Given a scalar time seriesx(t ) the reconstructed state space uses coordinates y1(t )= x(t )y2(t ) = dx(t )dt ym(t ) = dmx(t )dtm Here m + 1 is theembedding dimension chosen large enough that the dynamic inthe reconstructed state space is deterministic An alternative is totake successive time delays in x(t ) (ref 33) Using these methodsthe strange attractor hypothesis was eventually verified34

It is a short step once one has reconstructed the state spaceunderlying a chaotic signal to determine whether you can alsoextract the equations of motion themselves That is does the signaltell you which differential equations it obeys The answer is yes35This sound works quite well if and this will be familiar onehas made the right choice of representation for the lsquoright-handsidersquo of the differential equations Should one use polynomialFourier or wavelet basis functions or an artificial neural netGuess the right representation and estimating the equations ofmotion reduces to statistical quadrature parameter estimationand a search to find the lowest embedding dimension Guesswrong though and there is little or no clue about how toupdate your choice

The answer to this conundrum became the starting point for analternative approach to complexitymdashonemore suitable for physicalsystems The answer is articulated in computational mechanics36an extension of statistical mechanics that describes not only asystemrsquos statistical properties but also how it stores and processesinformationmdashhow it computes

The theory begins simply by focusing on predicting a time seriesXminus2Xminus1X0X1 In the most general setting a prediction is adistribution Pr(Xt |xt ) of futures Xt = XtXt+1Xt+2 conditionedon a particular past xt = xtminus3xtminus2xtminus1 Given these conditionaldistributions one can predict everything that is predictableabout the system

At root extracting a processrsquos representation is a very straight-forward notion do not distinguish histories that make the samepredictions Once we group histories in this way the groups them-selves capture the relevant information for predicting the futureThis leads directly to the central definition of a processrsquos effectivestates They are determined by the equivalence relation

xt sim xt primehArrPr(Xt |xt )=Pr(Xt |xt prime)

The equivalence classes of the relation sim are the processrsquoscausal states Smdashliterally its reconstructed state space and theinduced state-to-state transitions are the processrsquos dynamic T mdashitsequations of motion Together the statesS and dynamic T give theprocessrsquos so-called ε-machine

Why should one use the ε-machine representation of aprocess First there are three optimality theorems that say itcaptures all of the processrsquos properties36ndash38 prediction a processrsquosε-machine is its optimal predictor minimality compared withall other optimal predictors a processrsquos ε-machine is its minimalrepresentation uniqueness any minimal optimal predictor isequivalent to the ε-machine

Second we can immediately (and accurately) calculate thesystemrsquos degree of randomness That is the Shannon entropy rateis given directly in terms of the ε-machine

hmicro=minussumσisinS

Pr(σ )sumx

Pr(x|σ )log2Pr(x|σ )

where Pr(σ ) is the distribution over causal states and Pr(x|σ ) is theprobability of transitioning from state σ onmeasurement x

Third the ε-machine gives us a new propertymdashthe statisticalcomplexitymdashand it too is directly calculated from the ε-machine

Cmicro=minussumσisinS

Pr(σ )log2Pr(σ )

The units are bits This is the amount of information the processstores in its causal states

Fourth perhaps the most important property is that theε-machine gives all of a processrsquos patterns The ε-machine itselfmdashstates plus dynamicmdashgives the symmetries and regularities ofthe system Mathematically it forms a semi-group39 Just asgroups characterize the exact symmetries in a system theε-machine captures those and also lsquopartialrsquo or noisy symmetries

Finally there is one more unique improvement the statisticalcomplexity makes over KolmogorovndashChaitin complexity theoryThe statistical complexity has an essential kind of representationalindependence The causal equivalence relation in effect extractsthe representation from a processrsquos behaviour Causal equivalencecan be applied to any class of systemmdashcontinuous quantumstochastic or discrete

Independence from selecting a representation achieves theintuitive goal of using UTMs in algorithmic information theorymdashthe choice that in the end was the latterrsquos undoing Theε-machine does not suffer from the latterrsquos problems In this sensecomputational mechanics is less subjective than any lsquocomplexityrsquotheory that per force chooses a particular representational scheme

To summarize the statistical complexity defined in terms of theε-machine solves the main problems of the KolmogorovndashChaitincomplexity by being representation independent constructive thecomplexity of an ensemble and ameasure of structure

In these ways the ε-machine gives a baseline against whichany measures of complexity or modelling in general should becompared It is a minimal sufficient statistic38

To address one remaining question let us make explicit theconnection between the deterministic complexity framework andthat of computational mechanics and its statistical complexityConsider realizations x` from a given information source Breakthe minimal UTM program P for each into two componentsone that does not change call it the lsquomodelrsquo M and one thatdoes change from input to input E the lsquorandomrsquo bits notgenerated by M Then an objectrsquos lsquosophisticationrsquo is the lengthof M (refs 4041)

SOPH(x`)= argmin|M | P =M+Ex`=UTM P

10|H 05|H05|T

05|T05|H10|T

Figure 1 | ε-machines for four information sources a The all-headsprocess is modelled with a single state and a single transition Thetransition is labelled p|x where pisin [01] is the probability of the transitionand x is the symbol emitted b The fair-coin process is also modelled by asingle state but with two transitions each chosen with equal probabilityc The period-2 process is perhaps surprisingly more involved It has threestates and several transitions d The uncountable set of causal states for ageneric four-state HMM The causal states here are distributionsPr(ABCD) over the HMMrsquos internal states and so are plotted as points ina 4-simplex spanned by the vectors that give each state unit probabilityPanel d reproduced with permission from ref 44 copy 1994 Elsevier

As done with the KolmogorovndashChaitin complexity we candefine the ensemble-averaged sophistication 〈SOPH〉 of lsquotypicalrsquorealizations generated by the source The result is that the averagesophistication of an information source is proportional to itsprocessrsquos statistical complexity42

KC(`)propCmicro+hmicro`That is 〈SOPH〉propCmicro

Notice how far we come in computational mechanics bypositing only the causal equivalence relation From it alone wederive many of the desired sometimes assumed features of othercomplexity frameworks We have a canonical representationalscheme It is minimal and so Ockhamrsquos razor43 is a consequencenot an assumption We capture a systemrsquos pattern in the algebraicstructure of the ε-machine We define randomness as a processrsquosε-machine Shannon-entropy rate We define the amount oforganization in a process with its ε-machinersquos statistical complexityIn addition we also see how the framework of deterministiccomplexity relates to computational mechanics

ApplicationsLet us address the question of usefulness of the foregoingby way of examples

Letrsquos start with the Prediction Game an interactive pedagogicaltool that intuitively introduces the basic ideas of statisticalcomplexity and how it differs from randomness The first steppresents a data sample usually a binary times series The second askssomeone to predict the future on the basis of that data The finalstep asks someone to posit a state-based model of the mechanismthat generated the data

The first data set to consider is x0 = HHHHHHHmdashtheall-heads process The answer to the prediction question comesto mind immediately the future will be all Hs x =HHHHHSimilarly a guess at a state-based model of the generatingmechanism is also easy It is a single state with a transitionlabelled with the output symbol H (Fig 1a) A simple modelfor a simple process The process is exactly predictable hmicro = 0

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

Figure 2 | Structure versus randomness a In the period-doubling route to chaos b In the two-dimensional Ising-spinsystem Reproduced with permissionfrom a ref 36 copy 1989 APS b ref 61 copy 2008 AIP

bits per symbol Furthermore it is not complex it has vanishingcomplexity Cmicro= 0 bits

The second data set is for example x0 = THTHTTHTHHWhat I have done here is simply flip a coin several times and reportthe results Shifting frombeing confident and perhaps slightly boredwith the previous example people take notice and spend a good dealmore time pondering the data than in the first case

The prediction question now brings up a number of issues Onecannot exactly predict the future At best one will be right onlyhalf of the time Therefore a legitimate prediction is simply to giveanother series of flips from a fair coin In terms of monitoringonly errors in prediction one could also respond with a series ofall Hs Trivially right half the time too However this answer getsother properties wrong such as the simple facts that Ts occur andoccur in equal number

The answer to the modelling question helps articulate theseissues with predicting (Fig 1b) The model has a single statenow with two transitions one labelled with a T and one withan H They are taken with equal probability There are severalpoints to emphasize Unlike the all-heads process this one ismaximally unpredictable hmicro = 1 bitsymbol Like the all-headsprocess though it is simple Cmicro= 0 bits again Note that the modelis minimal One cannot remove a single lsquocomponentrsquo state ortransition and still do prediction The fair coin is an example of anindependent identically distributed process For all independentidentically distributed processesCmicro=0 bits

In the third example the past data are x0 = HTHTHTHTHThis is the period-2 process Prediction is relatively easy once onehas discerned the repeated template word w =TH The predictionis x = THTHTHTH The subtlety now comes in answering themodelling question (Fig 1c)

There are three causal states This requires some explanationThe state at the top has a double circle This indicates that it is a startstatemdashthe state in which the process starts or from an observerrsquospoint of view the state in which the observer is before it beginsmeasuring We see that its outgoing transitions are chosen withequal probability and so on the first step a T or an H is producedwith equal likelihood An observer has no ability to predict whichThat is initially it looks like the fair-coin process The observerreceives 1 bit of information In this case once this start state is leftit is never visited again It is a transient causal state

Beyond the first measurement though the lsquophasersquo of theperiod-2 oscillation is determined and the process has movedinto its two recurrent causal states If an H occurred then it

is in state A and a T will be produced next with probability1 Conversely if a T was generated it is in state B and thenan H will be generated From this point forward the processis exactly predictable hmicro = 0 bits per symbol In contrast to thefirst two cases it is a structurally complex process Cmicro= 1 bitConditioning on histories of increasing length gives the distinctfuture conditional distributions corresponding to these threestates Generally for p-periodic processes hmicro = 0 bits symbolminus1

and Cmicro= log2p bitsFinally Fig 1d gives the ε-machine for a process generated

by a generic hidden-Markov model (HMM) This example helpsdispel the impression given by the Prediction Game examplesthat ε-machines are merely stochastic finite-state machines Thisexample shows that there can be a fractional dimension set of causalstates It also illustrates the general case for HMMs The statisticalcomplexity diverges and so we measure its rate of divergencemdashthecausal statesrsquo information dimension44

As a second example let us consider a concrete experimentalapplication of computational mechanics to one of the venerablefields of twentieth-century physicsmdashcrystallography how to findstructure in disordered materials The possibility of turbulentcrystals had been proposed a number of years ago by Ruelle53Using the ε-machine we recently reduced this idea to practice bydeveloping a crystallography for complexmaterials54ndash57

Describing the structure of solidsmdashsimply meaning theplacement of atoms in (say) a crystalmdashis essential to a detailedunderstanding of material properties Crystallography has longused the sharp Bragg peaks in X-ray diffraction spectra to infercrystal structure For those cases where there is diffuse scatteringhowever findingmdashlet alone describingmdashthe structure of a solidhas been more difficult58 Indeed it is known that without theassumption of crystallinity the inference problem has no uniquesolution59 Moreover diffuse scattering implies that a solidrsquosstructure deviates from strict crystallinity Such deviations cancome in many formsmdashSchottky defects substitution impuritiesline dislocations and planar disorder to name a few

The application of computational mechanics solved thelongstanding problemmdashdetermining structural information fordisordered materials from their diffraction spectramdashfor the specialcase of planar disorder in close-packed structures in polytypes60The solution provides the most complete statistical descriptionof the disorder and from it one could estimate the minimumeffective memory length for stacking sequences in close-packedstructures This approach was contrasted with the so-called fault

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

Figure 3 | Complexityndashentropy diagrams a The one-dimensional spin-12 antiferromagnetic Ising model with nearest- and next-nearest-neighbourinteractions Reproduced with permission from ref 61 copy 2008 AIP b Complexityndashentropy pairs (hmicroCmicro) for all topological binary-alphabetε-machines with n= 16 states For details see refs 61 and 63

model by comparing the structures inferred using both approacheson two previously published zinc sulphide diffraction spectra Thenet result was that having an operational concept of pattern led to apredictive theory of structure in disorderedmaterials

As a further example let us explore the nature of the interplaybetween randomness and structure across a range of processesAs a direct way to address this let us examine two families ofcontrolled systemmdashsystems that exhibit phase transitions Considerthe randomness and structure in two now-familiar systems onefrom nonlinear dynamicsmdashthe period-doubling route to chaosand the other from statistical mechanicsmdashthe two-dimensionalIsing-spin model The results are shown in the complexityndashentropydiagrams of Fig 2 They plot a measure of complexity (Cmicro and E)versus the randomness (H (16)16 and hmicro respectively)

One conclusion is that in these two families at least the intrinsiccomputational capacity is maximized at a phase transition theonset of chaos and the critical temperature The occurrence of thisbehaviour in such prototype systems led a number of researchersto conjecture that this was a universal interdependence betweenrandomness and structure For quite some time in fact therewas hope that there was a single universal complexityndashentropyfunctionmdashcoined the lsquoedge of chaosrsquo (but consider the issues raisedin ref 62) We now know that although this may occur in particularclasses of system it is not universal

It turned out though that the general situation is much moreinteresting61 Complexityndashentropy diagrams for two other processfamilies are given in Fig 3 These are rather less universal lookingThe diversity of complexityndashentropy behaviours might seem toindicate an unhelpful level of complication However we now seethat this is quite useful The conclusion is that there is a widerange of intrinsic computation available to nature to exploit andavailable to us to engineer

Finally let us return to address Andersonrsquos proposal for naturersquosorganizational hierarchy The idea was that a new lsquohigherrsquo level isbuilt out of properties that emerge from a relatively lsquolowerrsquo levelrsquosbehaviour He was particularly interested to emphasize that the newlevel had a new lsquophysicsrsquo not present at lower levels However whatis a lsquolevelrsquo and how different should a higher level be from a lowerone to be seen as new

We can address these questions now having a concrete notion ofstructure captured by the ε-machine and a way to measure it thestatistical complexityCmicro In line with the theme so far let us answerthese seemingly abstract questions by example In turns out thatwe already saw an example of hierarchy when discussing intrinsiccomputational at phase transitions

Specifically higher-level computation emerges at the onsetof chaos through period-doublingmdasha countably infinite stateε-machine42mdashat the peak of Cmicro in Fig 2a

How is this hierarchical We answer this using a generalizationof the causal equivalence relation The lowest level of description isthe raw behaviour of the system at the onset of chaos Appealing tosymbolic dynamics64 this is completely described by an infinitelylong binary string We move to a new level when we attempt todetermine its ε-machine We find at this lsquostatersquo level a countablyinfinite number of causal states Although faithful representationsmodels with an infinite number of components are not onlycumbersome but not insightful The solution is to apply causalequivalence yet againmdashto the ε-machinersquos causal states themselvesThis produces a new model consisting of lsquometa-causal statesrsquothat predicts the behaviour of the causal states themselves Thisprocedure is called hierarchical ε-machine reconstruction45 and itleads to a finite representationmdasha nested-stack automaton42 Fromthis representation we can directly calculate many properties thatappear at the onset of chaos

Notice though that in this prescription the statistical complexityat the lsquostatersquo level diverges Careful reflection shows that thisalso occurred in going from the raw symbol data which werean infinite non-repeating string (of binary lsquomeasurement statesrsquo)to the causal states Conversely in the case of an infinitelyrepeated block there is no need to move up to the level of causalstates At the period-doubling onset of chaos the behaviour isaperiodic although not chaotic The descriptional complexity (theε-machine) diverged in size and that forced us to move up to themeta- ε-machine level

This supports a general principle that makes Andersonrsquos notionof hierarchy operational the different scales in the natural world aredelineated by a succession of divergences in statistical complexityof lower levels On the mathematical side this is reflected in thefact that hierarchical ε-machine reconstruction induces its ownhierarchy of intrinsic computation45 the direct analogue of theChomsky hierarchy in discrete computation theory65

Closing remarksStepping back one sees that many domains face the confoundingproblems of detecting randomness and pattern I argued that thesetasks translate into measuring intrinsic computation in processesand that the answers give us insights into hownature computes

Causal equivalence can be adapted to process classes frommany domains These include discrete and continuous-outputHMMs (refs 456667) symbolic dynamics of chaotic systems45

molecular dynamics68 single-molecule spectroscopy6769 quantumdynamics70 dripping taps71 geomagnetic dynamics72 andspatiotemporal complexity found in cellular automata73ndash75 and inone- and two-dimensional spin systems7677 Even then there aremany remaining areas of application

Specialists in the areas of complex systems and measures ofcomplexity will miss a number of topics above more advancedanalyses of stored information intrinsic semantics irreversibilityand emergence46ndash52 the role of complexity in a wide range ofapplication fields including biological evolution78ndash83 and neuralinformation-processing systems84ndash86 to mention only two ofthe very interesting active application areas the emergence ofinformation flow in spatially extended and network systems7487ndash89the close relationship to the theory of statistical inference8590ndash95and the role of algorithms from modern machine learning fornonlinear modelling and estimating complexity measures Eachtopic is worthy of its own review Indeed the ideas discussed herehave engaged many minds for centuries A short and necessarilyfocused review such as this cannot comprehensively cite theliterature that has arisen even recently not so much for itssize as for its diversity

I argued that the contemporary fascination with complexitycontinues a long-lived research programme that goes back to theorigins of dynamical systems and the foundations of mathematicsover a century ago It also finds its roots in the first days ofcybernetics a half century ago I also showed that at its core thequestions its study entails bear on some of the most basic issues inthe sciences and in engineering spontaneous organization originsof randomness and emergence

The lessons are clear We now know that complexity arisesin a middle groundmdashoften at the orderndashdisorder border Naturalsystems that evolve with and learn from interaction with their im-mediate environment exhibit both structural order and dynamicalchaosOrder is the foundation of communication between elementsat any level of organization whether that refers to a population ofneurons bees or humans For an organismorder is the distillation ofregularities abstracted from observations An organismrsquos very formis a functional manifestation of its ancestorrsquos evolutionary and itsown developmental memories

A completely ordered universe however would be dead Chaosis necessary for life Behavioural diversity to take an example isfundamental to an organismrsquos survival No organism canmodel theenvironment in its entirety Approximation becomes essential toany system with finite resources Chaos as we now understand itis the dynamical mechanism by which nature develops constrainedand useful randomness From it follow diversity and the ability toanticipate the uncertain future

There is a tendency whose laws we are beginning tocomprehend for natural systems to balance order and chaos tomove to the interface between predictability and uncertainty Theresult is increased structural complexity This often appears asa change in a systemrsquos intrinsic computational capability Thepresent state of evolutionary progress indicates that one needsto go even further and postulate a force that drives in timetowards successively more sophisticated and qualitatively differentintrinsic computation We can look back to times in whichthere were no systems that attempted to model themselves aswe do now This is certainly one of the outstanding puzzles96how can lifeless and disorganized matter exhibit such a driveThe question goes to the heart of many disciplines rangingfrom philosophy and cognitive science to evolutionary anddevelopmental biology and particle astrophysics96 The dynamicsof chaos the appearance of pattern and organization andthe complexity quantified by computation will be inseparablecomponents in its resolution

Received 28 October 2011 accepted 30 November 2011published online 22 December 2011

References1 Press W H Flicker noises in astronomy and elsewhere Comment Astrophys

7 103ndash119 (1978)2 van der Pol B amp van der Mark J Frequency demultiplication Nature 120

363ndash364 (1927)3 Goroff D (ed) in H Poincareacute New Methods of Celestial Mechanics 1 Periodic

And Asymptotic Solutions (American Institute of Physics 1991)4 Goroff D (ed) H Poincareacute New Methods Of Celestial Mechanics 2

Approximations by Series (American Institute of Physics 1993)5 Goroff D (ed) in H Poincareacute New Methods Of Celestial Mechanics 3 Integral

Invariants and Asymptotic Properties of Certain Solutions (American Institute ofPhysics 1993)

6 Crutchfield J P Packard N H Farmer J D amp Shaw R S Chaos Sci Am255 46ndash57 (1986)

7 Binney J J Dowrick N J Fisher A J amp Newman M E J The Theory ofCritical Phenomena (Oxford Univ Press 1992)

8 Cross M C amp Hohenberg P C Pattern formation outside of equilibriumRev Mod Phys 65 851ndash1112 (1993)

9 Manneville P Dissipative Structures and Weak Turbulence (Academic 1990)10 Shannon C E A mathematical theory of communication Bell Syst Tech J

27 379ndash423 623ndash656 (1948)11 Cover T M amp Thomas J A Elements of Information Theory 2nd edn

(WileyndashInterscience 2006)12 Kolmogorov A N Entropy per unit time as a metric invariant of

automorphisms Dokl Akad Nauk SSSR 124 754ndash755 (1959)13 Sinai Ja G On the notion of entropy of a dynamical system

Dokl Akad Nauk SSSR 124 768ndash771 (1959)14 Anderson P W More is different Science 177 393ndash396 (1972)15 Turing A M On computable numbers with an application to the

Entscheidungsproblem Proc Lond Math Soc 2 42 230ndash265 (1936)16 Solomonoff R J A formal theory of inductive inference Part I Inform Control

7 1ndash24 (1964)17 Solomonoff R J A formal theory of inductive inference Part II Inform Control

7 224ndash254 (1964)18 Minsky M L in Problems in the Biological Sciences Vol XIV (ed Bellman R

E) (Proceedings of Symposia in AppliedMathematics AmericanMathematicalSociety 1962)

19 Chaitin G On the length of programs for computing finite binary sequencesJ ACM 13 145ndash159 (1966)

20 Kolmogorov A N Three approaches to the concept of the amount ofinformation Probab Inform Trans 1 1ndash7 (1965)

21 Martin-Loumlf P The definition of random sequences Inform Control 9602ndash619 (1966)

22 Brudno A A Entropy and the complexity of the trajectories of a dynamicalsystem Trans Moscow Math Soc 44 127ndash151 (1983)

23 Zvonkin A K amp Levin L A The complexity of finite objects and thedevelopment of the concepts of information and randomness by means of thetheory of algorithms Russ Math Survey 25 83ndash124 (1970)

24 Chaitin G Algorithmic Information Theory (Cambridge Univ Press 1987)25 Li M amp Vitanyi P M B An Introduction to Kolmogorov Complexity and its

Applications (Springer 1993)26 Rissanen J Universal coding information prediction and estimation

IEEE Trans Inform Theory IT-30 629ndash636 (1984)27 Rissanen J Complexity of strings in the class of Markov sources IEEE Trans

Inform Theory IT-32 526ndash532 (1986)28 Blum L Shub M amp Smale S On a theory of computation over the real

numbers NP-completeness Recursive Functions and Universal MachinesBull Am Math Soc 21 1ndash46 (1989)

29 Moore C Recursion theory on the reals and continuous-time computationTheor Comput Sci 162 23ndash44 (1996)

30 Shannon C E Communication theory of secrecy systems Bell Syst Tech J 28656ndash715 (1949)

31 Ruelle D amp Takens F On the nature of turbulence Comm Math Phys 20167ndash192 (1974)

32 Packard N H Crutchfield J P Farmer J D amp Shaw R S Geometry from atime series Phys Rev Lett 45 712ndash716 (1980)

33 Takens F in Symposium on Dynamical Systems and Turbulence Vol 898(eds Rand D A amp Young L S) 366ndash381 (Springer 1981)

34 Brandstater A et al Low-dimensional chaos in a hydrodynamic systemPhys Rev Lett 51 1442ndash1445 (1983)

35 Crutchfield J P amp McNamara B S Equations of motion from a data seriesComplex Syst 1 417ndash452 (1987)

36 Crutchfield J P amp Young K Inferring statistical complexity Phys Rev Lett63 105ndash108 (1989)

37 Crutchfield J P amp Shalizi C R Thermodynamic depth of causal statesObjective complexity via minimal representations Phys Rev E 59275ndash283 (1999)

38 Shalizi C R amp Crutchfield J P Computational mechanics Pattern andprediction structure and simplicity J Stat Phys 104 817ndash879 (2001)

39 Young K The Grammar and Statistical Mechanics of Complex Physical SystemsPhD thesis Univ California (1991)

40 Koppel M Complexity depth and sophistication Complexity 11087ndash1091 (1987)

41 Koppel M amp Atlan H An almost machine-independent theory ofprogram-length complexity sophistication and induction InformationSciences 56 23ndash33 (1991)

42 Crutchfield J P amp Young K in Entropy Complexity and the Physics ofInformation Vol VIII (ed Zurek W) 223ndash269 (SFI Studies in the Sciences ofComplexity Addison-Wesley 1990)

43 William of Ockham Philosophical Writings A Selection Translated with anIntroduction (ed Philotheus Boehner O F M) (Bobbs-Merrill 1964)

44 Farmer J D Information dimension and the probabilistic structure of chaosZ Naturf 37a 1304ndash1325 (1982)

45 Crutchfield J P The calculi of emergence Computation dynamics andinduction Physica D 75 11ndash54 (1994)

46 Crutchfield J P in Complexity Metaphors Models and Reality Vol XIX(eds Cowan G Pines D amp Melzner D) 479ndash497 (Santa Fe Institute Studiesin the Sciences of Complexity Addison-Wesley 1994)

47 Crutchfield J P amp Feldman D P Regularities unseen randomness observedLevels of entropy convergence Chaos 13 25ndash54 (2003)

48 Mahoney J R Ellison C J James R G amp Crutchfield J P How hidden arehidden processes A primer on crypticity and entropy convergence Chaos 21037112 (2011)

49 Ellison C J Mahoney J R James R G Crutchfield J P amp Reichardt JInformation symmetries in irreversible processes Chaos 21 037107 (2011)

50 Crutchfield J P in Nonlinear Modeling and Forecasting Vol XII (eds CasdagliM amp Eubank S) 317ndash359 (Santa Fe Institute Studies in the Sciences ofComplexity Addison-Wesley 1992)

51 Crutchfield J P Ellison C J amp Mahoney J R Timersquos barbed arrowIrreversibility crypticity and stored information Phys Rev Lett 103094101 (2009)

52 Ellison C J Mahoney J R amp Crutchfield J P Prediction retrodictionand the amount of information stored in the present J Stat Phys 1361005ndash1034 (2009)

53 Ruelle D Do turbulent crystals exist Physica A 113 619ndash623 (1982)54 Varn D P Canright G S amp Crutchfield J P Discovering planar disorder

in close-packed structures from X-ray diffraction Beyond the fault modelPhys Rev B 66 174110 (2002)

55 Varn D P amp Crutchfield J P From finite to infinite range order via annealingThe causal architecture of deformation faulting in annealed close-packedcrystals Phys Lett A 234 299ndash307 (2004)

56 Varn D P Canright G S amp Crutchfield J P Inferring Pattern and Disorderin Close-Packed Structures from X-ray Diffraction Studies Part I ε-machineSpectral Reconstruction Theory Santa Fe Institute Working Paper03-03-021 (2002)

57 Varn D P Canright G S amp Crutchfield J P Inferring pattern and disorderin close-packed structures via ε-machine reconstruction theory Structure andintrinsic computation in Zinc Sulphide Acta Cryst B 63 169ndash182 (2002)

58 Welberry T R Diffuse x-ray scattering andmodels of disorder Rep Prog Phys48 1543ndash1593 (1985)

59 Guinier A X-Ray Diffraction in Crystals Imperfect Crystals and AmorphousBodies (W H Freeman 1963)

60 Sebastian M T amp Krishna P Random Non-Random and Periodic Faulting inCrystals (Gordon and Breach Science Publishers 1994)

61 Feldman D P McTague C S amp Crutchfield J P The organization ofintrinsic computation Complexity-entropy diagrams and the diversity ofnatural information processing Chaos 18 043106 (2008)

62 Mitchell M Hraber P amp Crutchfield J P Revisiting the edge of chaosEvolving cellular automata to perform computations Complex Syst 789ndash130 (1993)

63 Johnson B D Crutchfield J P Ellison C J amp McTague C S EnumeratingFinitary Processes Santa Fe Institute Working Paper 10-11-027 (2010)

64 Lind D amp Marcus B An Introduction to Symbolic Dynamics and Coding(Cambridge Univ Press 1995)

65 Hopcroft J E amp Ullman J D Introduction to Automata Theory Languagesand Computation (Addison-Wesley 1979)

66 Upper D R Theory and Algorithms for Hidden Markov Models and GeneralizedHidden Markov Models PhD thesis Univ California (1997)

67 Kelly D Dillingham M Hudson A amp Wiesner K Inferring hidden Markovmodels from noisy time sequences A method to alleviate degeneracy inmolecular dynamics Preprint at httparxivorgabs10112969 (2010)

68 Ryabov V amp Nerukh D Computational mechanics of molecular systemsQuantifying high-dimensional dynamics by distribution of Poincareacute recurrencetimes Chaos 21 037113 (2011)

69 Li C-B Yang H amp Komatsuzaki T Multiscale complex network of proteinconformational fluctuations in single-molecule time series Proc Natl AcadSci USA 105 536ndash541 (2008)

70 Crutchfield J P amp Wiesner K Intrinsic quantum computation Phys Lett A372 375ndash380 (2006)

71 Goncalves W M Pinto R D Sartorelli J C amp de Oliveira M J Inferringstatistical complexity in the dripping faucet experiment Physica A 257385ndash389 (1998)

72 Clarke R W Freeman M P amp Watkins N W The application ofcomputational mechanics to the analysis of geomagnetic data Phys Rev E 67160ndash203 (2003)

73 Crutchfield J P amp Hanson J E Turbulent pattern bases for cellular automataPhysica D 69 279ndash301 (1993)

74 Hanson J E amp Crutchfield J P Computational mechanics of cellularautomata An example Physica D 103 169ndash189 (1997)

75 Shalizi C R Shalizi K L amp Haslinger R Quantifying self-organization withoptimal predictors Phys Rev Lett 93 118701 (2004)

76 Crutchfield J P amp Feldman D P Statistical complexity of simpleone-dimensional spin systems Phys Rev E 55 239Rndash1243R (1997)

77 Feldman D P amp Crutchfield J P Structural information in two-dimensionalpatterns Entropy convergence and excess entropy Phys Rev E 67051103 (2003)

78 Bonner J T The Evolution of Complexity by Means of Natural Selection(Princeton Univ Press 1988)

79 Eigen M Natural selection A phase transition Biophys Chem 85101ndash123 (2000)

80 Adami C What is complexity BioEssays 24 1085ndash1094 (2002)81 Frenken K Innovation Evolution and Complexity Theory (Edward Elgar

Publishing 2005)82 McShea D W The evolution of complexity without natural

selectionmdashA possible large-scale trend of the fourth kind Paleobiology 31146ndash156 (2005)

83 Krakauer D Darwinian demons evolutionary complexity and informationmaximization Chaos 21 037111 (2011)

84 Tononi G Edelman G M amp Sporns O Complexity and coherencyIntegrating information in the brain Trends Cogn Sci 2 474ndash484 (1998)

85 BialekW Nemenman I amp Tishby N Predictability complexity and learningNeural Comput 13 2409ndash2463 (2001)

86 Sporns O Chialvo D R Kaiser M amp Hilgetag C C Organizationdevelopment and function of complex brain networks Trends Cogn Sci 8418ndash425 (2004)

87 Crutchfield J P amp Mitchell M The evolution of emergent computationProc Natl Acad Sci USA 92 10742ndash10746 (1995)

88 Lizier J Prokopenko M amp Zomaya A Information modification and particlecollisions in distributed computation Chaos 20 037109 (2010)

89 Flecker B Alford W Beggs J M Williams P L amp Beer R DPartial information decomposition as a spatiotemporal filter Chaos 21037104 (2011)

90 Rissanen J Stochastic Complexity in Statistical Inquiry(World Scientific 1989)

91 Balasubramanian V Statistical inference Occamrsquos razor and statisticalmechanics on the space of probability distributions Neural Comput 9349ndash368 (1997)

92 Glymour C amp Cooper G F (eds) in Computation Causation and Discovery(AAAI Press 1999)

93 Shalizi C R Shalizi K L amp Crutchfield J P Pattern Discovery in Time SeriesPart I Theory Algorithm Analysis and Convergence Santa Fe Institute WorkingPaper 02-10-060 (2002)

94 MacKay D J C Information Theory Inference and Learning Algorithms(Cambridge Univ Press 2003)

95 Still S Crutchfield J P amp Ellison C J Optimal causal inference Chaos 20037111 (2007)

96 Wheeler J A in Entropy Complexity and the Physics of Informationvolume VIII (ed Zurek W) (SFI Studies in the Sciences of ComplexityAddison-Wesley 1990)

AcknowledgementsI thank the Santa Fe Institute and the Redwood Center for Theoretical NeuroscienceUniversity of California Berkeley for their hospitality during a sabbatical visit

Additional informationThe author declares no competing financial interests Reprints and permissionsinformation is available online at httpwwwnaturecomreprints

Communities modules and large-scale structurein networksM E J Newman

Networks also called graphs by mathematicians provide a useful abstraction of the structure of many complex systemsranging from social systems and computer networks to biological networks and the state spaces of physical systems In thepast decade there have been significant advances in experiments to determine the topological structure of networked systemsbut there remain substantial challenges in extracting scientific understanding from the large quantities of data produced bythe experiments A variety of basic measures and metrics are available that can tell us about small-scale structure in networkssuch as correlations connections and recurrent patterns but it is considerably more difficult to quantify structure on mediumand large scales to understand the lsquobig picturersquo Important progress has been made however within the past few years aselection of which is reviewed here

A network is in its simplest form a collection of dots joinedtogether in pairs by lines (Fig 1) In the jargon of the fielda dot is called a lsquonodersquo or lsquovertexrsquo (plural lsquoverticesrsquo) and a

line is called an lsquoedgersquo Networks are used in many branches ofscience as a way to represent the patterns of connections betweenthe components of complex systems1ndash6 Examples include theInternet78 in which the nodes are computers and the edges are dataconnections such as optical-fibre cables food webs in biology910in which the nodes are species in an ecosystem and the edgesrepresent predatorndashprey interactions and social networks1112 inwhich the nodes are people and the edges represent any of avariety of different types of social interaction including friendshipcollaboration business relationships or others

In the past decade there has been a surge of interest in both em-pirical studies of networks13 and development of mathematical andcomputational tools for extracting insight from network data1ndash6One common approach to the study of networks is to focus onthe properties of individual nodes or small groups of nodes askingquestions such as lsquoWhich is the most important node in this net-workrsquo or lsquoWhich are the strongest connectionsrsquo Such approacheshowever tell us little about large-scale network structure It is thislarge-scale structure that is the topic of this paper

The best-studied form of large-scale structure in networks ismodular or community structure1415 A community in this contextis a dense subnetwork within a larger network such as a close-knitgroup of friends in a social network or a group of interlinked webpages on the World Wide Web (Fig 1) Although communitiesare not the only interesting form of large-scale structuremdashthereare others that we will come tomdashthey serve as a good illustrationof the nature and scope of present research in this area and willbe our primary focus

Communities are of interest for a number of reasons Theyhave intrinsic interest because they may correspond to functionalunits within a networked system an example of the kind oflink between structure and function that drives much of thepresent excitement about networks In a metabolic network16for instancemdashthe network of chemical reactions within a cellmdashacommunity might correspond to a circuit pathway or motif thatcarries out a certain function such as synthesizing or regulating avital chemical product17 In a social network a community mightcorrespond to an actual community in the conventional sense of the

Department of Physics and Center for the Study of Complex Systems University of Michigan Ann Arbor Michigan 48109 USA e-mail mejnumichedu

Figure 1 | Example network showing community structure The nodes ofthis network are divided into three groups with most connections fallingwithin groups and only a few between groups

word a group of people brought together by a common interest acommon location or workplace or family ties18

However there is another reason less often emphasized whya knowledge of community structure can be useful In manynetworks it is found that the properties of individual communitiescan be quite different Consider for example Fig 2 which showsa network of collaborations among a group of scientists at aresearch institute The network divides into distinct communities asindicated by the colours of the nodes (We will see shortly how thisdivision is accomplished) In this case the communities correspondclosely to the acknowledged research groups within the institute ademonstration that indeed the discovery of communities can pointto functional divisions in a system However notice also that thestructural features of the different communities are widely varyingThe communities highlighted in red and light blue for instanceappear to be loose-knit groups of collaborators working togetherin various combinations whereas the groups in yellow and darkblue are both organized around a central hub perhaps a group

Figure 2 |A network of collaborations among scientists at a researchinstitute Nodes in this network represent the scientists and there is anedge between any pair of scientists who co-authored a published paperduring the years of the study Colours represent communities asdiscovered using a modularity-maximization technique

leader or principal investigator of some kind Distinctions such asthese which may be crucial for understanding the behaviour ofthe system become apparent only when one looks at structure onthe community level

The network in this particular example has the nice property thatit is small enough and sparse enough to be drawn clearly on the pageOne does not need any calculations to pick out the communities inthis case a good eye will do the job However when we are workingwith larger or denser networks networks that can have thousandsor even millions of nodes (or a smaller number of nodes but verymany edges) clear visualization becomes impossible and we mustturn instead to algorithmic methods for community detection andthe development of such methods has been a highly active area ofresearch in the past few years15

The community-detection problem is challenging in part be-cause it is not verywell posed It is agreed that the basic problem is tofind locally dense regions in a network but this is not a precise for-mulation If one is to create a method for detecting communities inamechanical way onemust first define exactly what onemeans by acommunity Researchers have been aware of this issue from the out-set and have proposed a wide variety of definitions based on countsof edges within and between communities counts of paths acrossnetworks spectral properties of network matrices information-theoretic measures randomwalks andmany other quantities Withthis array of definitions comes a corresponding array of algorithmsthat seek to find the communities so defined141519ndash31 Unfortu-nately it is no easy matter to determine which of these algorithmsare the best because the perception of good performance itselfdepends on how one defines a community and each algorithmis necessarily good at finding communities according to its own

definition To get around this circularity we typically take one oftwo approaches In the first algorithms are tested against real-worldnetworks for which there is an accepted division into communitiesoften based on additionalmeasurements that are independent of thenetwork itself such as interviews with participants in a social net-work or analysis of the text of web pages If an algorithm can reliablyfind the accepted structure then it is considered successful In thesecond approach algorithms are tested against computer-generatednetworks that have some form of community structure artificiallyembedded within them A number of standard benchmark net-works have been proposed for this purpose such as the lsquofour groupsrsquonetworks14 or so-called the LFR benchmark networks32 A numberof studies have been published that compare the performance ofproposed algorithms in these benchmark tests3334 Although theseapproaches do set concrete targets for performance of community-detectionmethods there is room for debate over whether those tar-gets necessarily align with good performance in broader real-worldsituations If we tune our algorithms to solve specific benchmarkproblems we run the risk of creating algorithms that solve thoseproblemswell but other (perhapsmore realistic) problems poorly

This is a crucial issue and one that is worth bearing inmind as wetake a look in the following sections at the present state of researchon community detection As we will see however researchers havein spite of the difficulties come up with a range of approaches thatreturn real useful information about the large-scale structure ofnetworks and in the process have learned much both about indi-vidual networks that have been analysed and about mathematicalmethods for representing and understanding network structure

Hierarchical clusteringStudies of communities in networks go back at least to the 1970swhen a number of techniques were developed for their detectionparticularly in computer science and sociology In computerscience the problem of graph partitioning35 which is similarbut not identical to the problem of community detection hasreceived attention for its engineering applications but the methodsdeveloped such as spectral partitioning36 and the KernighanndashLin algorithm37 have also been fruitfully applied in other areasHowever it is thework of sociologists that is perhaps themost directancestor ofmodern techniques of community detection

An early and still widely used technique for detectingcommunities in social networks is hierarchical clustering511Hierarchical clustering is in fact not a single technique but anentire family of techniques with a single central principle if wecan derive a measure of how strongly nodes in a network areconnected together then by grouping the most strongly connectedwe can divide the network into communities Specific hierarchicalclusteringmethods differ on the particularmeasure of strength usedand on the rules by which we group strongly connected nodesMost common among themeasures used are the so-called structuralequivalence measures which focus on the number nij of commonnetwork neighbours that two nodes i j have In a social networkof friendships for example two people with many mutual friendsare more likely to be close than two people with few and thus acount of mutual friends can be used as a measure of connectionstrength Rather than using the raw count nij however one typicallynormalizes it in some way leading to measures such as the Jaccardcoefficient and cosine similarity For example the cosine similarityσij between nodes i and j is defined by

σij =nijradickikj

where ki is the degree of node i (that is the number of con-nections it has) This measure has the nice property that its

Figure 3 |Average-linkage clustering of a small social network This tree or lsquodendrogramrsquo shows the results of the application of average-linkagehierarchical clustering using cosine similarity to the well-known karate-club network of Zachary38 which represents friendship between members of auniversity sports club The calculation finds two principal communities in this case (the left and right subtrees of the dendrogram) which correspondexactly to known factions within the club (represented by the colours)

value falls always between zero and onemdashzero if the nodes haveno common neighbours and one if they have all their neigh-bours in common

Once one has defined a measure of connection strength onecan begin to group nodes together which is done in hierarchicalfashion first grouping single nodes into small groups thengrouping those groups into larger groups and so forth There are anumber of methods by which this grouping can be carried out thethree common ones being the methods known as single-linkagecomplete-linkage and average-linkage clustering Single-linkageclustering is the most widely used by far primarily because it issimple to implement but in fact average-linkage clustering gener-ally gives superior results and is notmuch harder to implement

Figure 3 shows the result of applying average-linkage hierarchicalclustering based on cosine similarity to a famous network fromthe social networks literature Zacharyrsquos karate-club network38This network represents patterns of friendship between membersof a karate club at a US university compiled from observationsand interviews of the clubrsquos 34 members The network is ofparticular interest because during the study a dispute arose amongthe clubrsquos members over whether to raise club fees Unable toreconcile their differences the members of the club split intotwo factions with one faction departing to start a separate clubIt has been claimed repeatedly that by examining the patternof friendships depicted in the network (which was compiledbefore the split happened) one can predict the membership of thetwo factions1420262738ndash40

Figure 3 shows the output of the hierarchical clustering proce-dure in the form of a tree or lsquodendrogramrsquo representing the order inwhich nodes are grouped together into communities It should beread from the bottom up at the bottom we have individual nodesthat are grouped first into pairs and then into larger groups aswe move up the tree until we reach the top where all nodes havebeen gathered into one group In a single image this dendrogramcaptures the entire hierarchical clustering process Horizontal cutsthrough the figure represent the groups at intermediate stages

As we can see the method in this case joins the nodes togetherinto two large groups consisting of roughly half the network eachbefore finally joining those two into one group at the top of thedendrogram It turns out that these two groups correspondpreciselyto the groups into which the club split in real life which areindicated by the colours in the figure Thus in this case the methodworks well It has effectively predicted a future social phenomenonthe split of the club fromquantitative datameasured before the splitoccurred It is the promise of outcomes such as this that drivesmuchof the present interest in networks

Hierarchical clustering is straightforward to understand and toimplement but it does not always give satisfactory results As itexists in many variants (different strength measures and differentlinkage rules) and different variants give different results it is notclear which results are the lsquocorrectrsquo ones Moreover the methodhas a tendency to group together those nodes with the strongestconnections but leave out those with weaker connections so thatthe divisions it generates may not be clean divisions into groupsbut rather consist of a few dense cores surrounded by a periphery ofunattached nodes Ideally wewould like amore reliablemethod

Optimization methodsOver the past decade or so researchers in physics and appliedmathematics have taken an active interest in the community-detection problem and introduced a number of fruitful approachesAmong the first proposals were approaches based on a measureknown as betweenness142141 in which one calculates one ofseveral measures of the flow of (imaginary) traffic across theedges of a network and then removes from the network thoseedges with the most traffic Two other related approaches arethe use of fluid-flow19 and current-flow analogies42 to identifyedges for removal the latter idea has been revived recentlyto study structure in the very largest networks30 A differentclass of methods are those based on information-theoretic ideassuch as the minimum-description-length methods of Rosvall andBergstrom2643 and related methods based on statistical inferencesuch as the message-passing method of Hastings25 Another largeclass exploits links between community structure and processestaking place on networks such as randomwalks4445 Potts models46or oscillator synchronization47 A contrasting set of approachesfocuses on the detection of lsquolocal communitiesrsquo2324 and seeks toanswer the question of whether we can given a single nodeidentify the community to which it belongs without first findingall communities in the network In addition to being useful forstudying limited portions of larger networks this approach can giverise to overlapping communities in which a node can belong tomore than one community (The generalized community-detectionproblem in which overlaps are allowed in this way has been an areaof increasing interest within the field in recent years2231)

However the methods most heavily studied by physicists per-haps unsurprisingly are those that view the community-detectionproblem by analogy with equilibrium physical processes and treatit as an optimization task The basic idea is to define a quantitythat is high for lsquogoodrsquo divisions of a network and low for lsquobadrsquoones and then to search through possible divisions for the onewith the highest score This approach is similar to the minimization

of energy when finding the ground state or stable state of aphysical system and the connection has been widely exploited Avariety of different measures for assigning scores have been pro-posed such as the so-called EI ratio48 likelihood-based measures49and others50 but the most widely used is the measure knownas the modularity1851

Suppose you are given a network and a candidate division intocommunities A simple measure of the quality of that divisionis the fraction of edges that fall within (rather than between)communities If this fraction is high then you have a good division(Fig 1) However this measure is not ideal It is maximized byputting all nodes in a single group together which is a correct buttrivial form of community structure and not of particular interestA better measure is the so-called modularity which is defined to bethe fraction of edges within communities minus the expected valueof that fraction if the positions of the edges are randomized51 Ifthere are more edges within communities than one would find in arandomized network then the modularity will be positive and largepositive values indicate good community divisions

Let Aij be equal to the number of edges between nodes i and j(normally zero or one) Aij is an element of the lsquoadjacency matrixrsquoof the network It can be shown that for a network with m edgesin total the expected number that fall between nodes i and j ifthe positions of the edges are randomized is given by kikj2mwhere ki is again the degree of node i Thus the actual number ofedges between i and j minus the expected number is Aijminuskikj2mand the modularity Q is the sum of this quantity over all pairs ofnodes that fall in the same community If we label the communitiesand define si to be the label of the community to which node ibelongs then we can write

[Aijminus

kikj2m

]δsisj

where δij is the Kronecker delta and the leading constant 12m isincluded only by conventionmdashit normalizesQ to measure fractionsof edges rather than total numbers but its presence has no effect onthe position of the modularity maximum

The modularity takes precisely the form H = minussum

ij Jijδsisj ofthe Hamiltonian of a (disordered) Potts model apart from aminus sign and hence its maximization is equivalent to finding theground state of the Potts modelmdashthe community assignments si actsimilarly to spins on the nodes of the network Unfortunately directoptimization of the modularity by an exhaustive search through thepossible spin states is intractable for any but the smallest of net-works and faster indirect (but exact) algorithms have been provedrigorously not to exist52 A variety of approximate techniques fromphysics and elsewhere however are applicable to the problem andseem to give good but not perfect solutions with relatively modestcomputational effort These include simulated annealing1753greedy algorithms5455 semidefinite programming28 spectralmethods56 and several others4057 Modularity maximization formsthe basis for other more complex approaches as well such as themethodof Blondel et al27 amultiscalemethod inwhichmodularityis first optimized using a greedy local algorithm then a lsquosupernet-workrsquo is formed whose nodes represent the communities so discov-ered and the greedy algorithm is repeated on this supernetworkThe process iterates until no further improvements in modularityare possible This method has become widely used by virtue of itsrelative computational efficiency and the high quality of the resultsit returns In a recent comparative study it was found to be one of thebest available algorithms when tested against computer-generatedbenchmark problems of the type described in the introduction34

Figure 2 showing collaboration patterns among scientists is anexample of community detection using modularity maximization

One of the nice features of the modularity method is that one doesnot need to know in advance the number of communities containedin the network a free maximization of the modularity in whichthe number of communities is allowed to vary will tell us the mostadvantageous number as well as finding the exact division of thenodes among communities

Although modularity maximization is efficient widely usedand gives informative results itmdashlike hierarchical clusteringmdashhasdeficiencies In particular it has a known bias in the size of thecommunities it findsmdashit has a preference for communities of sizeroughly equal to the square root of the size of the network58Modifications of the method have been proposed that allow oneto vary this preferred size5960 but not to eliminate the preferencealtogether The modularity method also ignores any informationstored in the positions of edges that run between communitiesas modularity is calculated by counting only within-group edgesone could move the between-group edges around in any wayone pleased and the value of the modularity would not changeat all One might imagine that one could do a better job ofdetecting communities if one were to make use of the informationrepresented by these edges

In the past few years therefore researchers have started to lookfor a more principled approach to community detection and havegravitated towards the method of block modelling a method thattraces its roots back to the 1970s (refs 6162) but which has recentlyenjoyed renewed popularity with some powerful new methodsand results emerging

Block modelsBlock modelling63ndash67 is in effect a form of statistical inference fornetworks In the same way that we can gain some understandingfrom conventional numerical data by fitting say a straight linethrough data points so we can gain understanding of the structureof networks by fitting them to a statistical network model Inparticular if we are interested in community structure then we cancreate a model of networks that contain such structure then fit itto an observed network and in the process learn about communitystructure in that observed network if it exists

A simple example of a block model is a model network inwhich one has a certain number n of nodes and each node isassigned to one of several labelled groups or communities Inaddition one specifies a set of probabilities prs which representthe probability that there will be an edge between a node ingroup r and a node in group s This model can be used forinstance in a generative process to create a random network withcommunity structure By making the edge probabilities higher forpairs of nodes in the same group and lower for pairs in differentgroups then generating a set of edges independently with exactlythose probabilities one can produce an artificial network that hasmany edges within groups and few between themmdashthe classiccommunity structure

However we can also turn the experiment around and ask lsquoIf weobserve a real network and we suppose that it was generated by thismodel what would the values of the modelrsquos parameters have tobersquo More precisely what values of the parameters are most likelyto have generated the network we see in real life This leads us toa lsquomaximum likelihoodrsquo formulation of the community-detectionproblem The probability or likelihood that an observed networkwas generated by this blockmodel is given by

L=prodiltj

pAijsisj (1minuspsisj )

1minusAij

where Aij is an element of the adjacency matrix as beforeand si is again the community to which node i belongs Now

we simply maximize this quantity over the probabilities prs andthe communities si Again we have turned the detection ofcommunities into an optimization problem albeit a harder onethan the modularity-maximization problem The values of theprobabilities prs are usually of lesser interest to us but if we can findthe community parameters si that maximize the likelihood then wehave solved our community-detection problem

Although it seems elegant and well-founded in principle thesurprising thing about this approach at least as we have describedit here is that it does not work well Figure 4a shows an exampleapplication of (a slight variant of) the method to a network ofweblogs or lsquoblogsrsquomdashpersonal web pages maintained by individualsor groups on which they publish their thoughts on topics of theirchoosing This particular network which was assembled by Adamicand Glance68 is composed of blogs about US politics that wereactive around the time of the US presidential election in 2004 andthe edges in the network represent web hyperlinks between blogsAdamic and Glance showed that this network was strongly dividedinto two communities one of left-leaning (that is liberal) blogswhich commonly link to one another and the other of right-leaning(conservative) ones which also link to one another but that therewere few links between left and right The communities appear asroughly the left and right halves of the network as it is drawn inFig 4a The colours in the figure show the division of the networkinto two communities foundwith themaximum likelihoodmethodabove and it is clear that the method has failed to find the knowndivision in this case What has gone wrong

On closer inspection we find that the method fails in this casebecause it does not take into account the wide variation among thedegrees of nodes in the network In this network (and many others)degrees vary over a great range whereas degrees in the block modelare Poisson distributed and narrowly peaked about their meanThis means in effect that there is no choice of parameters for themodel that gives a good fit to the data Fitting this block modelis similar to fitting a straight line through an inherently curvedset of data pointsmdashyou can do it but it is unlikely to give you ameaningful answer

It turns out however that one can fix such problems by suitablymodifying the model Figure 4b shows a different fit to the samenetwork using now a lsquodegree-correctedrsquo block model that allows forwidely varying degrees49 As the figure shows the model now findsa division that corresponds closely to the known division betweenleft- and right-leaning blogs The moral of the story is that it is nothard to come up with models so unrealistic that they will not fitthe observed network for any parameter values and one must guardagainst this possibility if the method is to work

Once we deal with this issue however the block-model methodhas some promising features If we have found the parameter valuesfor the best fit of the model to an observed network we canthen plug those values back into the model and use the model togenerate further networks that are similar to the original networkbut not identical This ability to generate similar networks can beused for instance to guess at the locations of possible missingedges in a network For many networks our data are incompleteor unreliable and there may be edges missing from the recordedstructure Looking at a large selection of generated networks that aresimilar to the original one can find edges that appear often in thegenerated networks but not in the original such edges turn out tobe reliable candidates for missing data Guimeraacute and Sales-Pardo69have shown that this approach is at least as accurate as and oftenbetter than previousmethods for predictingmissing edges

Another nice feature of the block-model method is that it lendsitself to many variants that are suitable for particular types ofproblem For instance in some problems we can with some effortcarry out experiments to determine the community membership of

Figure 4 |Analysis of a network of links between web sites about USpolitics The two panels represent the divisions found in a network ofpolitical weblogs using two different versions of the block model methoda Division into two communities discovered using a fit to the basic blockmodel described in the text which fails to find the acknowledged division ofthe network into politically left- and right-leaning communities b Divisionusing a block model that corrects for the broad distribution of node degreesin the network This division corresponds closely to the acknowledged oneFigure reproduced with permission from ref 49 copy 2011 APS Network datataken from ref 68

a few nodes and the goal is to determine the rest In recent workYan et al70 have devised a variant of the block-model methodin which one can use the model to determine on which nodesthese experiments should be done by looking for the nodes whosemembership information will be most useful in the sense that itwill tell us as much as possible not only about the measured nodesbut also about the membership of other nodes in the network Theyshow that the accuracy of community detection can be enormouslyimproved by carrying out just a few experiments on nodes carefullychosen using this technique

However perhaps the most promising feature of the block-model method is that it is not limited to detecting traditionalcommunity structure in networks In principle any type ofstructure that can be formulated as a probabilistic model can bedetected including overlapping communities bipartite or k-partite

Figure 5 |Hierarchical divisions in a food web of grassland speciesOutlined sets of nodes represent groups of species at different levels in thehierarchy For clarity only two levels in the hierarchy are shown althoughfive levels were found in some parts of the network Reproduced fromref 71

structures communities within communities andmany others Thefield is only just beginning to explore the wide range of possibilitiesthat this approach offers but Fig 5 shows one example drawnfrom my own work71 In this study we examined the food web ofa grassland ecosystemmdashthe network of predatorndashprey interactionsbetween speciesmdashand searched for a generalized form of hierar-chical community structure in which groups divide into subgroupsand subsubgroups and so on Using a model that employs a treestructure reminiscent of the dendrogram of Fig 3 to represent thehierarchy of groups and edge probabilities that depend on shortestpaths through the tree we were able to discover an entire spectrumof structure within the network spanning the range from smallmotifs of a few nodes to the size of the entire network Of particularnote in this example is the way in which the method groups hostspecies (squares) with their parasites (yellow triangles) but at thenext level in the hierarchy also gathers the parasites separatelyinto their own groups In some sense the parasites have more incommon with each other than with their host and hence can bethought of as belonging to a separate group even though they haveno direct interactions with one another through the food web Thecalculation realizes this and divides the network accordingly

ConclusionThe study of network structure and its links with the function andbehaviour of complex systems is a large and active field of endeavorwith new results appearing daily and an energetic community ofresearchers working on both methods and applications Some ofthe ideas discussed here are now well established and widely usedwhereas others such as the block-modelmethods are being activelyresearched and developed and there are many others still that thereis not room to describe in this article The pace of developmentsis if anything accelerating and the field offers substantial promisefor those in physics biology the social sciences and elsewhere forwhom the ability to make sense of the structures large and smallfound in networks can open a new window on the behaviour ofsystems of many kinds

References1 Albert R amp Barabaacutesi A-L Statistical mechanics of complex networks

Rev Mod Phys 74 47ndash97 (2002)2 Dorogovtsev S N amp Mendes J F F Evolution of networks Adv Phys 51

1079ndash1187 (2002)3 Newman M E J The structure and function of complex networks SIAM Rev

45 167ndash256 (2003)4 Boccaletti S Latora V Moreno Y Chavez M amp Hwang D-U Complex

networks Structure and dynamics Phys Rep 424 175ndash308 (2006)5 Newman M E J Networks An Introduction (Oxford Univ Press 2010)6 Cohen R amp Havlin S Complex Networks Structure Stability and Function

(Cambridge Univ Press 2010)7 Faloutsos M Faloutsos P amp Faloutsos C On power-law relationships of the

internet topology Comput Commun Rev 29 251ndash262 (1999)8 Pastor-Satorras R amp Vespignani A Evolution and Structure of the Internet

(Cambridge Univ Press 2004)9 Pimm S L Food Webs 2nd edn (Univ Chicago Press 2002)10 Pascual M amp Dunne J A (eds) Ecological Networks Linking Structure to

Dynamics in Food Webs (Oxford Univ Press 2006)11 Wasserman S amp Faust K Social Network Analysis

(Cambridge Univ Press 1994)12 Scott J Social Network Analysis A Handbook 2nd edn (Sage 2000)13 Costa L da F Rodrigues F A Travieso G amp Boas P R V

Characterization of complex networks A survey of measurements Adv Phys56 167ndash242 (2007)

14 Girvan M amp Newman M E J Community structure in social and biologicalnetworks Proc Natl Acad Sci USA 99 7821ndash7826 (2002)

15 Fortunato S Community detection in graphs Phys Rep 486 75ndash174 (2010)16 Jeong H Tombor B Albert R Oltvai Z N amp Barabaacutesi A-L The large-scale

organization of metabolic networks Nature 407 651ndash654 (2000)17 Guimeragrave R amp Amaral L A N Functional cartography of complex metabolic

networks Nature 433 895ndash900 (2005)18 Newman M E J amp Girvan M Finding and evaluating community structure

in networks Phys Rev E 69 026113 (2004)19 Flake G W Lawrence S R Giles C L amp Coetzee F M Self-organization

and identification of Web communities IEEE Comput 35 66ndash71 (2002)20 Zhou H Distance dissimilarity index and network community structure

Phys Rev E 67 061901 (2003)21 Radicchi F Castellano C Cecconi F Loreto V amp Parisi D Defining

and identifying communities in networks Proc Natl Acad Sci USA 1012658ndash2663 (2004)

22 Palla G Dereacutenyi I Farkas I amp Vicsek T Uncovering the overlappingcommunity structure of complex networks in nature and society Nature 435814ndash818 (2005)

23 Bagrow J P amp Bollt E M Local method for detecting communitiesPhys Rev E 72 046108 (2005)

24 Clauset A Finding local community structure in networks Phys Rev E 72026132 (2005)

25 Hastings M B Community detection as an inference problem Phys Rev E74 035102 (2006)

26 Rosvall M amp Bergstrom C T An information-theoretic framework forresolving community structure in complex networks Proc Natl Acad Sci USA104 7327ndash7331 (2007)

27 Blondel V D Guillaume J-L Lambiotte R amp Lefebvre E Fast unfolding ofcommunities in large networks J Stat Mech 2008 P10008 (2008)

28 Agrawal G amp Kempe D Modularity-maximizing network communities viamathematical programming Eur Phys J B 66 409ndash418 (2008)

29 Hofman J M amp Wiggins C H Bayesian approach to network modularityPhys Rev Lett 100 258701 (2008)

30 Leskovec J Lang K Dasgupta A amp Mahoney M Community structurein large networks Natural cluster sizes and the absence of large well-definedclusters Internet Math 6 29ndash123 (2009)

31 Ahn Y-Y Bagrow J P amp Lehmann S Link communities reveal multiscalecomplexity in networks Nature 466 761ndash764 (2010)

32 Lancichinetti A Fortunato S amp Radicchi F Benchmark graphs for testingcommunity detection algorithms Phys Rev E 78 046110 (2008)

33 Danon L Duch J Diaz-Guilera A amp Arenas A Comparing communitystructure identification J Stat Mech P09008 (2005)

34 Lancichinetti A amp Fortunato S Community detection algorithms Acomparative analysis Phys Rev E 80 056117 (2009)

35 Schaeffer S E Graph clustering Comput Sci Rev 1 27ndash64 (2007)36 Pothen A Simon H amp Liou K-P Partitioning sparse matrices with

eigenvectors of graphs SIAM J Matrix Anal Appl 11 430ndash452 (1990)37 Kernighan B W amp Lin S An efficient heuristic procedure for partitioning

graphs Bell Syst Tech J 49 291ndash307 (1970)38 Zachary W W An information flow model for conflict and fission in small

groups J Anthropol Res 33 452ndash473 (1977)

39 White D R amp Harary F The cohesiveness of blocks in social networksConnectivity and conditional density Sociol Methodol 31 305ndash359 (2001)

40 Duch J amp Arenas A Community detection in complex networks usingextremal optimization Phys Rev E 72 027104 (2005)

41 Wilkinson D M amp Huberman B A A method for finding communities ofrelated genes Proc Natl Acad Sci USA 101 5241ndash5248 (2004)

42 Wu F amp Huberman B A Finding communities in linear time A physicsapproach Eur Phys J B 38 331ndash338 (2004)

43 Rosvall M amp Bergstrom C T Multilevel compression of random walkson networks reveals hierarchical organization in large integrated systemsPLoS One 6 e18209 (2011)

44 Zhou H amp Lipowsky R Network Brownian Motion A New Method to MeasureVertexndashVertex Proximity and to Identify Communities and Subcommunities1062ndash1069 (Lecture Notes in Computer Science Vol 3038 Springer 2004)

45 Pons P amp Latapy M Proc 20th International Symposium on Computer andInformation Sciences 284ndash293 (Lecture Notes in Computer Science Vol 3733Springer 2005)

46 Reichardt J amp Bornholdt S Detecting fuzzy community structures in complexnetworks with a Potts model Phys Rev Lett 93 218701 (2004)

47 Boccaletti S Ivanchenko M Latora V Pluchino A amp Rapisarda ADetection of complex networks modularity by dynamical clusteringPhys Rev E 75 045102 (2007)

48 Karckhardt D amp Stern R Informal networks and organizational crises Anexperimental simulation Soc Psychol Q 51 123ndash140 (1988)

49 Karrer B amp Newman M E J Stochastic blockmodels and communitystructure in networks Phys Rev E 83 016107 (2011)

50 Li Z Zhang S Wang R-S Zhang X-S amp Chen L Quantitative function forcommunity detection Phys Rev E 77 036109 (2008)

51 Newman M E J Mixing patterns in networks Phys Rev E 67 026126 (2003)52 Brandes U et al Proc 33rd International Workshop on Graph-Theoretic

Concepts in Computer Science (Lecture Notes in Computer ScienceVol 4769Springer 2007)

53 Medus A Acuntildea G amp Dorso C O Detection of community structures innetworks via global optimization Physica A 358 593ndash604 (2005)

54 Clauset A Newman M E J amp Moore C Finding community structure invery large networks Phys Rev E 70 066111 (2004)

55 Wakita K amp Tsurumi T in Proc IADIS International ConferenceWWWInternet 2007 (eds Isaiacuteas P Nunes M B amp Barroso J) 153ndash162(IADIS Press 2007)

56 Newman M E J Modularity and community structure in networksProc Natl Acad Sci USA 103 8577ndash8582 (2006)

57 Shuzhuo L Yinghui C Haifeng D amp Feldman M W A genetic algorithmwith local search strategy for improved detection of community structureComplexity 15 53ndash60 (2010)

58 Fortunato S amp Bartheacuteleacutemy M Resolution limit in community detectionProc Natl Acad Sci USA 104 36ndash41 (2007)

59 Reichardt J amp Bornholdt S Statistical mechanics of community detectionPhys Rev E 74 016110 (2006)

60 Arenas A Fernandez A amp Gomez S Analysis of the structureof complex networks at different resolution levels New J Phys 10053039 (2008)

61 Breiger R L Boorman S A amp Arabie P An algorithm for clusteringrelations data with applications to social network analysis and comparison withmultidimensional scaling J Math Psychol 12 328ndash383 (1975)

62 Holland P W Laskey K B amp Leinhardt S Stochastic blockmodels Somefirst steps Soc Networks 5 109ndash137 (1983)

63 Snijders T A B amp Nowicki K Estimation and prediction for stochasticblockmodels for graphs with latent block structure J Classification 1475ndash100 (1997)

64 Nowicki K amp Snijders T A B Estimation and prediction for stochasticblockstructures J Am Stat Assoc 96 1077ndash1087 (2001)

65 Airoldi E M Blei D M Fienberg S E amp Xing E P Mixed membershipstochastic blockmodels J Mach Learning Res 9 1981ndash2014 (2008)

66 Goldenberg A Zheng A X Feinberg S E amp Airoldi E MA survey of statistical network structures Found Trends Mach Learning 21ndash117 (2009)

67 Bickel P J amp Chen A A nonparametric view of network models andNewmanndashGirvan and other modularities Proc Natl Acad Sci USA 10621068ndash21073 (2009)

68 Adamic L A amp Glance N Proc WWW-2005 Workshop on the WebloggingEcosystem (2005)

69 Guimeragrave R amp Sales-Pardo M Missing and spurious interactions andthe reconstruction of complex networks Proc Natl Acad Sci USA 10622073ndash22078 (2009)

70 Yan X Zhu Y Rouquier J-B amp Moore C in Proc 17th ACM SIGKDDInternational Conference on Knowledge Discovery and Data Mining (Associationof Computing Machinery 2011)

71 Clauset A Moore C amp Newman M E J Hierarchical structure and theprediction of missing links in networks Nature 453 98ndash101 (2008)

AcknowledgementsSome of the work described here was financially supported by the US National ScienceFoundation under grants DMSndash0405348 and DMSndash0804778

REVIEW ARTICLES | INSIGHTPUBLISHED ONLINE 22 DECEMBER 2011 | DOI 101038NPHYS2160

Modelling dynamical processes in complexsocio-technical systemsAlessandro Vespignani12

In recent years the increasing availability of computer power and informatics tools has enabled the gathering of reliable dataquantifying the complexity of socio-technical systems Data-driven computational models have emerged as appropriate tools totackle the study of dynamical phenomena as diverse as epidemic outbreaks information spreading and Internet packet routingThese models aim at providing a rationale for understanding the emerging tipping points and nonlinear properties that oftenunderpin the most interesting characteristics of socio-technical systems Here using diffusion and contagion phenomena asprototypical examples we review some of the recent progress in modelling dynamical processes that integrates the complexfeatures and heterogeneities of real-world systems

Questions concerning how pathogens spread in populationnetworks how blackouts can spread on a nationwide scaleor how efficiently we can search and retrieve data on large

information structures are generally related to the dynamics ofspreading and diffusion processes Social behaviour the spreadof cultural norms or the emergence of consensus may oftenbe modelled as the dynamical interaction of a set of connectedagents Phenomena as diverse as ecosystems or animal and insectbehaviour can all be described as the dynamic behaviour ofcollections of coupled oscillators Although all these phenomenarefer to very different systems their mathematical descriptionrelies on very similar models that depend on the definitionand characterization of a large number of individuals and theirinteractions in spatially extended systems

The modelling of dynamical processes is a research field thatcrosses different disciplines and has developed an impressive arrayof methods and approaches ranging from simple explanatorymodels to realistic approaches capable of providing quantitativeinsight into real-world systems Initially these models usedsimplistic assumptions for the micro-processes of interaction andwere mostly concerned with the study of the emerging macro-levelbehaviour This interest has favoured the use of techniques akinto statistical physics and the analysis of nonlinear equilibriumand non-equilibrium physical systems in the study of collectivebehaviour in social and population systems In recent yearshowever the increase in interdisciplinary work and the availabilityof system-level high-quality data has opened the way to data-drivenmodels aimed at a realistic description of complex socio-technicalsystems Modelling approaches to dynamical processes in complexsystems have been expanded into schemes that explicitly includespatial structures and have thus grown into a multiscale frameworkin which the various possible granularities of the system areconsidered through different approximations These models offera number of interesting and sometimes unexpected behaviourswhose theoretical understanding represents a new challenge thathas considerably transformed the mathematical and conceptualframework for the study of dynamical processes in complex systems

Dynamical processes and phase transitionsThe study of dynamical processes and the emergence of macro-level collective behaviour in complex systems follows a conceptualroute essentially similar to the statistical physics approach to

1Department of Physics College of Computer and Information Sciences Bouveacute College of Health Sciences Northeastern University BostonMassachusetts 02115 USA 2Institute for Scientific Interchange (ISI) Torino 10133 Italy e-mail avespignanineuedu

non-equilibrium phase transitions A prototypical example is thatof contagion processes Epidemiologists computer scientists andsocial scientists share a common interest in studying contagionphenomena and rely on very similar spreading models forthe description of the diffusion of viruses knowledge andinnovations1ndash5 All these processes define a contagion dynamicsthat can be seen as an actual biological pathogen that spreadsfrom host to host or a piece of information or knowledge thatis transmitted during social interactions Let us consider thesimple susceptiblendashinfectedndashrecovered (SIR) epidemic model Inthis model infected individuals (labelled with the state I ) canpropagate the contagion to susceptible neighbours (labelled withthe state S) with rate λ while infected individuals recover withrate micro and become removed from the population This is theprototypical model for the spread of infectious diseases whereindividuals recover and are immune to disease after a typicaltime that on average can be expressed as the inverse of therecovery rate A classic variation of this model is the susceptiblendashinfectedndashsusceptible (SIS) model in which individuals revert tothe susceptible state with rate micro modelling the possibility ofre-infection of individuals The mapping between epidemic modelsand non-equilibrium phase transitions was pointed out in physicslong ago making those models of very broad relevance alsooutside the area of information and disease spreading The staticproperties of the SIR model can indeed be mapped to an edge-percolation process6 Analogously the SIS model can be regardedas a generalization of the contact-process model7 widely studiedas the paradigmatic example of an absorbing-state phase transitionwith a unique absorbing state8

A cornerstone feature of epidemic processes is the presence of theso-called epidemic threshold1 In a fully homogeneous populationthe behaviour of the SIR model is controlled by the reproductivenumber R0=βmicro where β = λ〈k〉 is the per-capita spreading ratewhich takes into account the average number of contacts 〈k〉 of eachindividual The reproductive number simply identifies the averagenumber of secondary cases generated by a primary case in anentirely susceptible population and defines an epidemic thresholdsuch that only if R0 ge 1 (β gemicro) can epidemics reach an endemicstate and spread into a closed population The SIS and SIR modelsare indeed characterized by a threshold defining the transitionbetween two very different regimes These regimes are determinedby the values of the disease parameters and characterized by

iinfin

1 β μ

Figure 1 | Phase diagram of epidemic models Illustration of the behaviourof the prevalence iinfin for the SIS and SIR model in a heterogeneous network(solid line) as a function of the spreading rate βmicro compared with thetheoretical prediction for a homogeneous network (dashed line) The figureclearly shows the difference between homogeneous and heterogeneousnetworks where the epidemic threshold is shifted to very small values Forscale-free networks with degree distribution exponent γ le 3 however theassociated prevalence iinfin is extremely small over a large range of values ofβmicro In other words as noted since the first work on epidemic spreading incomplex networks the bad news about the suppression (or very smallvalue) of the epidemic threshold is balanced by the very low prevalenceattained by the epidemic46

the global parameter iinfin which identifies the density of infectedindividuals (or nodes in a network) in the infinite-time limit Inthe limit of an infinitely large population this density is zerobelow the threshold and assumes a finite value above the thresholdFrom this perspective we can consider the epidemic threshold asthe critical point of the system and iinfin as representing the orderparameter characterizing the transition Below the critical point thesystem relaxes in a frozen state with null dynamicsmdashthe healthyphase Above this point a dynamical state characterized by amacroscopic number of infected individuals sets in defining aninfected phase (Fig 1)

Many other pioneering works in the area of social sciences usesimple dynamical models to explore the emergence of macro-levelcollective behaviour as a function of themicro-level processes actingamong the agents of a large population9ndash11 and the incursions bystatistical physicists in the area of social sciences have become veryfrequent (see for example the recent review by Castellano et al12)A first class of models is represented by behavioural models wherethe attributes of agents are binary variables similar to Ising spinsas in the case of the voter model13 the majority-rule model1415and the Sznajd model16 In other instances further realism hasbeen introduced by the use of continuous opinion variables17ndash19Along the path opened by Axelrod11 models in which opinions orcultures are represented by vectors of cultural traits have introducedthe notion of bounded confidence an agent will not interactwith any other agent independently of their opinions unless theopinions are close enough

Finally there is a vast class of models that focus on the analysisof diffusion processes as a tool to study phase transitions andemergent phenomena in simple models mimicking the routingof information packets in technological systems and networksIn this case the focus is on what lies behind the appearance ofcongestion and traffic self-similarity20ndash26 In traffic problems oneof the main issues is that the diffusion process is not randombut determined by recurrent patterns reinforcing mechanismsand routing strategies that represent formidable challenges to themodelling of systems27 Interestingly it is the study of trafficdynamics in the Internet and the World Wide Web that has madeclear the central role of networks and their structural propertiesin the understanding and characterization of dynamical processesin real-world systems

Box 1 | The heterogeneous mean-field approach

The heterogeneousmean-field approach generalizes for the caseof networks with arbitrary degree distribution the equationsdescribing the dynamical process by considering degree-blockvariables grouping nodes within the same degree class k If weconsider the SIS model the variables describing the system are ikand sk which respectively represent the fraction of nodes withdegree k in the infected and susceptible class The evolutionequation for the infected individual is

dik(t )dt=minusmicroik+λ[1minus ik(t )]k2k(t )

The first term just expresses the fact that any node in the infectedstate may recover with ratemicro The second term which generatesnew infected individuals is proportional to the probability oftransmission λ the degree k the probability 1minus ik that a vertexwith degree k is not infected and the density 2k of infectedneighbours of vertices of degree k which is the probabilityof contacting an infected individuals As we are still assuminga mean-field description of the system the latter term is theaverage probability that any given neighbour of a vertex ofdegree k is infected This quantity can be expressed as 2k(t )=sum

k prime P(kprime|k)ik prime(t ) which is the average over all possible degrees

k prime of the probability P(k prime|k) that any edge of a node of degree kis pointing to a node of degree k prime times the probability ik prime thatthe node is infected This expression can be further simplified byconsidering a random network in which the conditional proba-bility does not depend on the originating node In this case wehave that P(k prime|k)= k primeP(k prime)〈k〉 following simply from the factthat any edge has a probability proportional to the degree itselfof pointing to a node with degree k prime (see ref 38) On substitutingthe expression for 2 in the main equation and adopting theearly-epidemic assumption (that is assuming that all second-order terms of ik and rk can be neglected) we readily recover thetopology-dependent epidemic threshold result λmicro=〈k〉〈k2〉

Following the results obtained with the HMF assumption anumber of rigorous results that link the network topology tothe epidemic threshold have been derived535758 These resultsrelate the epidemic threshold to the largest eigenvalue of theadjacency matrix of the network showing that the HMF doesnot recover the correct behaviour for the SIS model when thedegree distribution of the graph P(k) sim kminusγ has γ gt 3 Therigorous results refer to quenched networks where the adjacencymatrix is fixed in time The HMF assumption instead in itsmean-field perspective is equivalent to a system in which edgesare continuously reshuffled so that the elements of the adjacencymatrix are defined by the effective probabilities kikj

sumiki that

two nodes i and j with degree ki and kj respectively areconnected This consideration clearly shows the shortcomings ofthe HMF assumption in the case of systems where the timescaleof the transmission or infection is very short with respect tothe duration of the contact and the adjacency matrix can beconsidered as quenched The HMF can be considered howeveras a description of the system closer to reality in situations wherethe transmission occurs on rapidly varying networks this is forinstance the case for many influenza-like illnesses where theinfectious period is much longer than the duration of contactsresponsible for the transmission57

Complex networks and dynamical processesWe live in an increasingly interconnected world where infras-tructures composed of different technological layers inter-operate

Figure 2 | Progression of an epidemic process The progression of asusceptiblendashinfected (SI) epidemic in a heavy-tailed network at threesnapshots of the process corresponding to time t= 5 10 and 20 measuredin unitary time integration steps of the model The SI model assumes thatinfected nodes will spread the infection indefinitely to neighbours with rateα In this case we know that the system is eventually completely infectedwhatever the spreading rate of the infection However we can highlight theeffect of topological fluctuations on the spreading hierarchy Susceptiblenodes are coloured blue and infected nodes are coloured from yellow to redaccording to the time of infection (red corresponding to later times) Thesize of a node is proportional to the node degree In general the first nodesto be infected are the large hubs with high degree then the epidemicprogresses in time by a dynamical cascade through degree classes finallyaffecting low-degree nodes

within the social component that drives their use and developmentExamples are the Internet the World Wide Web mobile tech-nologies and transportation and mobility infrastructures28ndash34 Themultiscale nature and complexity of these networks are crucialfeatures in understanding and managing socio-technical systemsand the dynamical processes occurring on top of them For thisreason in the past decade the study of models unfolding oncomplex networks has generated a body of work that includesresults of conceptual and practical relevance35ndash40 The resilience ofnetworks their vulnerability to attacks and their synchronizationproperties are all drastically affected by topological heterogeneitiesConsensus formation disease spreading and the accessibility ofinformation can benefit or be impaired by the connectivity patternof the population or infrastructure we are looking at Networkscience has thus become pervasive in the study of complex sys-tems and presented us with a number of surprising discoveries

that have steered our way of thinking on dynamical processes insocio-technical systems

One of the most important features affecting dynamicalprocesses in real-world networks is the presence of dynamicself-organization and the lack of characteristic scalesmdashtypicalhallmarks of complex systems40ndash44 Although those characteristicshave long been acknowledged as a relevant factor in determiningthe properties of dynamical processes many real-world networksexhibit levels of heterogeneity that were not anticipated until afew years ago In particular the various statistical distributionscharacterizing these networks are generally heavy-tailed skewedand varying over several orders of magnitude This is a verypeculiar feature typical of many natural and artificial complexnetworks characterized by virtually infinite degree fluctuationswhere the degree k of a given node represents its number ofconnections to other nodes In contrast to regular lattices andhomogeneous graphs characterized by nodes having a typicaldegree k close to the average 〈k〉 such networks are structured ina hierarchy where a few nodes (the hubs) have very high degreewhereas the vast majority of nodes have lower degrees This featureis usually manifest in a heavy-tailed degree distribution oftenapproximated by a power-law behaviour of the form P(k)sim kminusγ which implies a non-negligible probability of finding verticeswith very high degree4042ndash44 Furthermore the presence of large-scale fluctuations associated with heavy-tail distributions is alsoobserved for the intensity carried by the connecting links transportflows and other basic quantities that go beyond the connectivitydescription of the network45

The presence of large-scale fluctuations virtually acting at allscales of the network connectivity pattern calls for a mathematicalanalysis where the variables characterizing each node of the networkexplicitly enter the description of the system Unfortunately thegeneral solution handling the master equation of the system ishardly if ever achievablemdasheven for very simple dynamical pro-cesses For this reason a viable theoretical approach has to be basedon techniques such as mean-field and deterministic continuumapproximations which usually provide the understanding of thebasic phenomenology and phase diagram of the process understudy In both cases the heterogeneous nature of the network-connectivity pattern is introduced by aggregating variables accord-ing to a degree-block formalism that assumes that all nodes withthe same degree k are statistically equivalent384647 This assumptionallows the grouping of nodes in degree classes yielding a convenientrepresentation of the system For instance if for each node iwe associate a corresponding state σi characterizing its dynamicalstate a convenient representation of the system is provided by thequantity Sk which indicates the number of nodes of degree k in thedynamical state σ = s and the corresponding degree-block densityof nodes of degree k in the state s

sk =SkVk

where Vk is the number of nodes of degree k Finally the globalaverages on the network are given by the expression

ρs=sumk

P(k)sk

where ρs is the probability that any given node is in the state s Thisformalism defines a mean-field approximation within each degreeclass relaxing however the overall homogeneity assumption onthe degree distribution38 This framework first introduced for thedescription of epidemic processes is at the basis of the heteroge-neous mean-field (HMF) approach that allows the analytical studyof dynamical processes in complex networks by writing mean-fielddynamical equations for each degree class variable An example

Box 2 | The particlendashnetwork framework

The particlendashnetwork framework extends the HMF approach tothe case of a reactionndashdiffusion system in which particles (orindividuals) diffuse on a network with arbitrary topology Aconvenient representation of the system is therefore provided byquantities defined in terms of the degree k

Nk =1Vk

sumi|ki=k

where Vk is the number of nodes with degree k and the sumsrun over all nodes i having degree ki equal to k The degree-blockvariable Nk represents the average number of particles in nodeswith degree k The use of the HMF approach amounts to theassumption that nodes with degree k and thus the particles inthose nodes are statistically equivalent In this approximation thedynamics of particles randomly diffusing on the network is givenby a mean-field dynamical equation expressing the variation intime of the particle subpopulations Nk(t ) in each degree block kThis can simply be written as

partNk

partt=minusdkNk(t )+k

sumk prime

P(k prime|k)dk primekNk prime(t )

The first term of the equation just considers that only a fractionof particles dk moves out of the node per unit time The secondterm accounts for particles diffusing from its neighbours into thenode of degree k This term is proportional to the number oflinks k times the average number of particles coming from eachneighbour The number of particles arriving from each neighbouris thus equal to that of particles dk primekNk prime(t ) diffusing on any edgeconnecting a node of degree k prime with a node of degree k averagedover the conditional probability P(k prime|k) that an edge belonging toa node of degree k is pointing to a node of degree k prime Here the termdk primek is the diffusion rate along the edges connecting nodes of degreek and k prime The rate at which individuals leave a subpopulationwith degree k is then given by dk = k

sumk primeP(k

prime|k)dkk prime The function

P(k prime|k) encodes the topological connectivity properties of thenetwork and allows the study of different topologies and mixingpatterns The above equation explicitly introduces the diffusionof particles into the description of the system The equationcan easily be generalized to particles with different states andreacting among themselves by adding a reaction term to theabove equations For instance the generalization of the SIRmodeldescribed in the main text would consider three types of particledenoting infected susceptible and recovered individuals Thereaction taking place among individuals in the same node wouldbe the usual contagion process among susceptibles and infectedindividuals and the spontaneous recovery of infected individuals

The analysis of a simple diffusion process immediately indi-cates the importance of network topology In a random networkwith arbitrary degree distribution the stationary state reached bya swarm of particles diffusing with the same diffusive rate yieldsNk sim k and the probability to find a single diffusing walker in anode of degree k is

pk =k〈k〉

where V is the total number of nodes in the network Thisexpression implies that the higher the degree of the nodesthe greater the probability to be visited by the walker Thisobservation has profound consequences for the way we candiscover retrieve and rank information in complex networksThe PageRank algorithm117 is in this respect a major break-through based on the idea that a viable ranking depends onthe topological structure of the network and is defined byessentially simulating the random surfing process on the webgraph The most important pages are simply those with thehighest probability of being discovered if the web-surfer hadinfinite time to explore the web Analogously search processescan take advantage of this property using degree-biased searchingalgorithms that bias the routing of messages towards nodes withhigh degree115116

of the HMF approach is given in Box 1 for the case of the SISmodel The HMF technique is often the first line of attack towardsunderstanding the effects of complex connectivity patterns ondynamical processes and it has been used widely in a broad range ofphenomena although with different names and specific assump-tions depending on the problem at hand Although it containsseveral approximations the HMF approach readily shows that theheterogeneity found in the connectivity pattern of many networksmay drastically affect the unfolding of the dynamical process

The classic example for the effect of degree heterogeneity ondynamical processes in complex networks is epidemic spreadingThe previously discussed result of the presence of an epidemicthreshold in the SIR and SIS models is obtained under theassumption that each individual in the system has to a firstapproximation the same number of connections k〈k〉 Howeversocial heterogeneity and the existence of lsquosuper-spreadersrsquo have longbeen known in the epidemics literature48 Generally it is possible toshow that the reproductive rateR0 is renormalized by fluctuations inthe transmissibility or contact pattern as R0rarrR0(1+ f (ν)) wheref (ν) is a positive and increasing function of the standard deviationν of the individual transmissibility or connectivity pattern49 Inparticular by generalizing the dynamical equations of the SISmodel the HMF approach yields that the disease will affect afinite fraction of the population only if βmicro ge 〈k〉2〈k2〉 that is

the ratio between the first and second moments of the degreedistribution384647 This readily suggests that the topology of thenetwork enters the very definition of the epidemic thresholdFurthermore this implies that in heavy-tailed networks such that〈k2〉 rarrinfin in the limit of infinite network size we have a nullepidemic threshold Although this is not the case in any finite-sizereal-world network5051 larger heterogeneity levels lead to smallerepidemic thresholds (Fig 1) This is an important result whichindicates that heterogeneous networks behave very differently fromhomogeneous networks with respect to physical and dynamicalprocesses Indeed the heterogeneous connectivity pattern ofnetworks affects also the dynamical progression of the epidemicprocess which results in a striking hierarchical dynamics inwhich the infection propagates from higher-degree to lower-degreeclasses The infection first takes control of the high-degree verticesin the network then rapidly invades the network via a cascadethrough progressively lower-degree classes (Fig 2) It also turnsout that the time behaviour of epidemic outbreaks and the growthof the number of infected individuals are governed by a timescaleτ proportional to the ratio between the first and second momentof the networkrsquos degree distribution thus suggesting a velocity ofprogression that increaseswith the heterogeneity of the network52

The change of framework suggested by the network heterogene-ity in the case of epidemic processes has triggered many studies

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

Figure 3 | Illustration of the global threshold in reactionndashdiffusion processes a Schematic of the simplified modelling framework based on theparticlendashnetwork scheme At the macroscopic level the system is composed of a heterogeneous network of subpopulations The contagion processin one subpopulation (marked in red) can spread to other subpopulations as particles diffuse across subpopulations b At the microscopic leveleach subpopulation contains a population of individuals The dynamical process for instance a contagion phenomena is described by a simplecompartmentalization (compartments are indicated by different coloured dots) Within each subpopulation individuals can mix homogeneously oraccording to a subnetwork and can diffuse with rate d from one subpopulation to another following the edges of the network c A critical value dc of thediffusion strength for individuals or particles identifies a phase transition between a regime in which the contagion affects a large fraction of the systemand one in which only a small fraction is affected (see the discussion in the text) Panels a and b reproduced from ref 118

aimed at providing a more rigorous analytical basis for the resultsobtained with the HMF and other approximate methods exploringdifferent spreading models53ndash58 Equally important is the researchactivity concerned with developing dynamical ad hoc strategies fornetwork protection targeted immunization strategies and targetedprophylaxis that evolve with time might be particularly effectivein the control of epidemics on heterogeneous patterns comparedwith massive uniform vaccinations or stationary interventions59ndash62Following the results on epidemic processes an avalanche of studiesaddressed the study of the effect of the networkrsquos structure on thebehaviour of the most widely used classes of dynamical processesFor instance in the area of synchronization it has been shownthat networks with heavy-tailed degree distributions and thereforea large number of hubs are more difficult to synchronize thanhomogeneous networks a counterintuitive insight dubbed theparadox of heterogeneity63ndash66 In the case of packet-traffic routinghomogeneous networks have typically much larger congestionthresholds than heterogeneous graphs67ndash69 Finally a wealth ofsurprising results often overturning the commonwisdom obtainedby studies on regular networks have been harvested on the voterand the Axelrod models70ndash73 and many other models for theemergence of cooperation3874

Reactionndashdiffusion processes and computational thinkingAlthough most approaches assume systems in which each nodeof the network corresponds to a single individual it is of crucialimportance for the study of many phenomena to provide a generalunderstanding of processes where the multiple occupancy of nodesis a key feature Examples of multiple occupancy are provided bychemical reactions in which different molecules or atoms diffusein space and may react whenever in close contact Mechanisticmetapopulation epidemic models where particles represent peoplemoving between different locations and the routing of information

packets in technological networks provides relevant examples in thecase of socio-technical systems75ndash79 All those phenomena fall intothe category of reactionndashdiffusion processes where each node i isallowed to have any non-negative integer number of particles Niso that the total particle population of the system is N =

The particlendashnetwork framework extends the heterogeneous mean-field approach to reactionndashdiffusion systems in networks witharbitrary degree distribution (Box 2) Particles diffuse along theedges connecting nodes with a diffusion coefficient that depends onthe node degree andor other nodesrsquo attributes Within each nodeparticles may react according to different schemes characterizingthe interaction dynamic of the system

The consideration of complex networks in reactionndashdiffusionsystems has broadened our knowledge of non-equilibriumreactionndashdiffusion systems in heterogeneous systems For instancethe Turing mechanism represents a classical model for theformation of self-organized spatial structures in non-equilibriumactivatorndashinhibitor systems By studying the Turingmechanism80 insystems with heterogeneous connectivity patterns it has been foundthat the relevant instabilities of the systems are localized in a setof vertices with degree inversely proportional to the characteristicscale of diffusion81 Interestingly and contrary to other models andsystems where the hubs are the playmakers the segregation processtakes place mainly in vertices of low degree

Another interesting example is that of simple epidemic pro-cesses such as the SIR model in a metapopulation context7982ndash90In this case each node of the network is a subpopulation (ideally anurban area) connected by a transportation system (the edges of thenetwork) that allows individuals to move from one subpopulationto another (Fig 3) If we assume a diffusion rate d for each individ-ual and consider that the single-population reproductive numberof the SIR model is R0 gt 1 we can easily identify two differentlimits If d = 0 any epidemic occurring in a given subpopulation

will remain confined no individual could travel to a differentsubpopulation and spread the infection across the system In thelimit drarrinfin we have that individuals are constantly wanderingfrom one subpopulation to the other and the system is in practiceequivalent to awell-mixed unique population In this case asR0gt1the epidemic will spread across the entire system A transitionpoint between these two regimes occurs at a threshold value dcof the diffusion rate identifying a global invasion threshold Thisthreshold cannot be uncovered by continuousmodels as it is relatedto the stochastic diffusion rate of single individuals Furthermorethe global invasion threshold is affected by the connectivity fluctu-ations of the metapopulation network In particular the greater thenetwork heterogeneity the smaller the value of the diffusion rateabove which the epidemic may globally invade the metapopulationsystem This result assumes a particular relevance as it explainswhy travel restrictions seem to be highly ineffective in containingepidemics the complexity and heterogeneity of present-day trans-port networks favour considerably the global spread of infectiousdiseases Only infeasibly tight mobility restrictions reducing globaltravel fluxes by 90 ormore would be effective849192

Reactionndashdiffusion models lend themselves to the implemen-tation of large-scale computer simulations (Monte-Carlo andindividual-based simulations) that allow one to track microscop-ically the state of each node and the evolution of the dynamicalprocess At the most detailed level the introduction of agent-basedmodels has enabled the usual modelling perspective to be extendedfurther by simulating the population and embedding environmenton an individual-by-individual basis An example is epidemic mod-elling where spatially structured and agent-basedmodels at variousgranularities (country inter-city intra-city) have been pushed tothe computational limits with the integration of huge amountof data describing the flows of people andor animals93ndash97 Thesemodels can generate results at an unprecedented level of detail andhave been used successfully in the analysis and anticipation of realepidemics such as the 2009 H1N1 pandemic9899 Computer simu-lations thus become valuable in allowing both in silico experimentsthat would be infeasible in real systems and the capability to analyseand forecast scenarios This computational approach is also helpingto guide researchers in identifying typical nonlinear behaviourand tipping points100 not accessible by analytical means using thenumerical simulations as a novel experimental workbench101102

Co-evolution timescale and controlAlthough in recent years our understanding of dynamical processesin complex networks has progressed at an exponential pace thereare still a number of major challenges that keep the researchcommunity actively engaged The first challenge stems from thefact that the analysis of dynamical processes is generally performedin the presence of a timescale separation between the networkevolution and the dynamical process unfolding on its structureIn one limit we can consider the network as quenched in itsconnectivity pattern thus evolving on a timescale that is muchlonger that the dynamical process itself In the other limiting casethe network evolves on a timescalemuch shorter than the dynamicalprocess which thus effectively disappears from the definition ofthe interaction among individuals such that this interaction canbe conveniently replaced by effective random coupling Althoughthe timescale separation is extremely convenient with a view tothe numerical and analytical tractability of the models networksgenerally evolve on a timescale that might be comparable to thatof the dynamical process Furthermore the network propertiesused in defining models generally represent a time-integratedstatic snapshot of the system However in many systems thetiming and duration of interactions define processes on a timescalevery different from and often conflicting with those of the

Figure 4 |Visualization of the dynamical network generated by Twitterinteractions Twitter is a microblogging tool that allows users to post andrelay (rsquore-tweetrsquo) short messages The topic of the message is signalled byshort identifiers (mentions hash-tags and urls) This feature allows oneto trace the spreading of specific discussion topics (also called memes)The figure shows the diffusion network for the tag gop Each nodecorresponds to an individual user Blue edges represent re-tweets andorange edges represent mentions Two communities are clearly visiblecorresponding to politically left- and right-leaning users113Communications between the two communities take place primarilythrough the use of mentions while within a group communication occursthrough re-tweets The figure obtained using the Truthy infrastructure114clearly exemplifies the co-evolution of the communication network with thespreading process

time-integrated view This highlights the importance of consideringthe concurrency of network evolution and dynamical processes inrealisticmodels to avoidmisleading conclusions103ndash106

A second challenge is the co-evolution of networks with thedynamical process Access to the mathematical and statistical lawsthat characterize the interplay and feedback mechanisms betweenthe network evolution and the dynamical processes is extremelyimportant especially in social systems where the adaptive natureof agents is of paramount importance106ndash108 The spreading of anopinion is affected by the interaction among individuals but thepresence andor establishment of interaction among individuals isaffected by their opinion This issue is increasingly relevant in thearea of the modern social networks populating the information-technology ecosystem such as those defined by the Facebook andTwitter applications In this case the network and the spread ofinformation cannot be defined in isolation because of rapidlychanging interactions and modes of communication that dependon the type of information exchanged and the adaptive behaviourof individuals (Fig 4)

The adaptive behaviour of individuals to the dynamicalprocesses they are involved in represents another modellingchallenge as it calls for the understanding of the feedbackamong different and competing dynamical processes For instancerelatively little systematic work has been done to provide coupledbehaviourndashdisease models able to close the feedback loop between

behavioural changes triggered in the population by an individualrsquosperception of the disease spread and the actual disease spread109110Similar issues arise in many areas where we find competingprocesses of adaptation and awareness to information or knowledgespreading in a population111

Finally the overall goal is not only to understand complexsystems mathematically describe their structure and dynamicsand predict their behaviour but also to control their dynamicsAlso in this case although control theory offers a large set ofmathematical tools for steering engineered and natural systems weare just taking the first steps towards a full understanding of how thenetwork heterogeneities influence our ability to control the networkdynamics andhow the network evolution impacts controllability112

ConclusionsThere are no doubts that a complete understanding of complexsocio-technical systems requires diving into the specifics of eachsystem by adopting a domain-specific perspective Data-drivenmodels however are generating new questions the answers towhich should preferably be analytical and applicable to a wide rangeof systems What are the fundamental limits to predictability withcomputational modelling How does our understanding dependon the level of accuracy of our description and knowledge of thestate of the system The research community needs nowmore thanever the kind of basic theoretical understanding that would helpdiscriminate betweenwhat is relevant andwhat is superfluous in thedescription of socio-technical systems This is a crucial endeavour ifwe want to complement data-driven approaches with a conceptualunderstanding that would help guide the management predictionand control of dynamical processes in complex systemsmdashaconceptual understanding that necessarily descends from the studyof the dynamicalmodels and processes presented here

References1 Keeling M J amp Rohani P Modeling Infectious Diseases in Humans and

Animals (Princeton Univ Press 2008)2 Goffman W amp Newill V A Generalization of epidemic theory An

application to the transmission of ideas Nature 204 225ndash228 (1964)3 Rapoport A Spread of information through a population with

socio-structural bias I Assumption of transitivity Bull Math Biol 15523ndash533 (1953)

4 Tabah A N Literature dynamics Studies on growth diffusion andepidemics Annu Rev Inform Sci Technol 34 249ndash286 (1999)

5 Lloyd A L amp May R M How viruses spread among computers and peopleScience 292 1316ndash1317 (2001)

6 Grassberger P On the critical behavior of the general epidemic process anddynamical percolationMath Biosci 63 157ndash172 (1983)

7 Harris T E Contact interactions on a lattice Ann Prob 2 969ndash988 (1974)8 Marro J amp Dickman R Nonequilibrium Phase Transitions in Lattice Models

(Cambridge Univ Press 1999)9 Granovetter M Threshold models of collective behavior Am J Sociol 83

1420ndash1443 (1978)10 Nowak A Szamrej J amp Lataneacute B From private attitude to public opinion

A dynamic theory of social impact Psychol Rev 97 362ndash376 (1990)11 Axelrod R The Complexity of Cooperation (Princeton Univ Press 1997)12 Castellano C Fortunato S amp Loreto V Statistical physics of social dynamics

Rev Mod Phys 81 591ndash646 (2009)13 Krapivsky P L Kinetics of monomerndashmonomer surface catalytic reactions

Phys Rev A 45 1067ndash1072 (1992)14 Galam S Minority opinion spreading in random geometry Eur Phys J B 25

403ndash406 (2002)15 Krapivsky P L amp Redner S Dynamics of majority rule in two-state

interacting spin systems Phys Rev Lett 90 238701 (2003)16 Sznajd-Weron K amp Sznajd J Opinion evolution in closed community

Int J Mod Phys C 11 1157ndash1165 (2000)17 Deffuant G Neau D Amblard F amp Weisbuch G Mixing beliefs among

interacting agents Adv Complex Syst 3 87ndash98 (2000)18 Hegselmann R amp Krause U Opinion dynamics and bounded confidence

models analysis and simulation J Art Soc Soc Sim 5 2 (2002)19 Ben-Naim E Krapivsky P L amp Redner S Bifurcations and patterns in

compromise processes Physica D 183 190ndash204 (2003)

20 Leland W E Taqqu M S Willinger W ampWilson D V On the self-similarnature of Ethernet traffic IEEEACM Trans Netw 2 1ndash15 (1994)

21 Csabai I 1f noise in computer network traffic J Phys A 27 L417ndashL42 (1994)22 Soleacute R V amp Valverde S Information transfer and phase transitions in a

model of internet traffic Physica A 289 595ndash605 (2001)23 Willinger W Govindan R Jamin S Paxson V amp Shenker S Scaling

phenomena in the Internet Critically examining criticality Proc Natl AcadSci USA 99 2573ndash2580 (2002)

24 Valverde S amp Soleacute R V Internetrsquos critical path horizon Eur Phys J B 38245ndash252 (2004)

25 Tadić B Thurner S amp Rodgers G J Traffic on complex networksTowards understanding global statistical properties from microscopic densityfluctuations Phys Rev E 69 036102 (2004)

26 Crovella M E amp Krishnamurthy B Internet Measurements InfrastructureTraffic and Applications (John Wiley 2006)

27 Helbing D Traffic and related self-driven many particle systemsRev Mod Phys 73 1067ndash1141 (2001)

28 Albert R Jeong H amp Barabaacutesi A-L Internet Diameter of the World-WideWeb Nature 401 130ndash131 (1999)

29 Pastor-Satorras R amp Vespignani A Evolution and Structure of the Internet AStatistical Physics Approach (Cambridge Univ Press 2004)

30 Brockmann D Hufnagel L amp Geisel T The scaling laws of human travelNature 439 462ndash465 (2006)

31 Onnela J-P et al Structure and tie strengths in mobile communicationnetworks Proc Natl Acad Sci USA 104 7332ndash7337 (2007)

32 Gonzaacutelez M C Hidalgo C A amp Barabaacutesi A-L Understanding individualhuman mobility patterns Nature 453 779ndash782 (2008)

33 Lazer D et al Life in the network The coming age of computational socialscience Science 323 721ndash723 (2009)

34 Vespignani A Predicting the behavior of tecno-social systems Science 325425ndash428 (2009)

35 Albert R amp Barabaacutesi A-L Statistical mechanics of complex networksRev Mod Phys 74 47ndash97 (2002)

36 Boccaletti S et al Complex networks Structure and dynamics Phys Rep424 175ndash308 (2006)

37 Dorogovtsev S N Goltsev A V amp Mendes J F F Critical phenomena incomplex networks Rev Mod Phys 80 1275ndash1335 (2008)

38 Barrat A Barthelemy M amp Vespignani A Dynamical Processes on ComplexNetworks (Cambridge Univ Press 2008)

39 Cohen R amp Havlin S Complex Networks Structure Robustness and Function(Cambridge Univ Press 2010)

40 Newman M E J Networks An Introduction (Oxford Univ Press 2010)41 Watts D J amp Strogatz S H Collective dynamics of lsquosmall-worldrsquo networks

Nature 393 440ndash442 (1998)42 Barabaacutesi A-L amp Albert R Emergence of scaling in random networks Science

286 509ndash512 (1999)43 Dorogovtsev S N amp Mendes J F F Evolution of Networks From Biological

Nets to the Internet and WWW (Oxford Univ Press 2003)44 Amaral L A N Scala A Barthlemy M amp Stanley H E Classes of

small-world networks Proc Natl Acad Sci USA 97 11149ndash11154 (2005)45 Barrat A Barthlemy M Pastor-Satorras R amp Vespignani A The

architecture of complex weighted networks Proc Natl Acad Sci USA 1013747ndash3752 (2004)

46 Pastor-Satorras R amp Vespignani A Epidemic spreading in scale-freenetworks Phys Rev Lett 86 3200ndash3203 (2001)

47 Moreno Y Pastor-Satorras R amp Vespignani A Epidemic outbreaks incomplex heterogeneous networks Eur Phys J B 26 521ndash529 (2002)

48 Hethcote H W amp Yorke J A Gonorrhea Transmission and controlLect Notes Biomath 56 1ndash105 (1984)

49 Anderson R M amp May R M Infectious Diseases in Humans (Oxford UnivPress 1992)

50 May R M amp Lloyd A L Infection dynamics on scale-free networksPhys Rev E 64 066112 (2001)

51 Pastor-Satorras R amp Vespignani R Epidemic dynamics in finite sizescale-free networks Phys Rev E 65 035108(R) (2002)

52 Barthelemy M Barrat A Pastor-Satorras R amp Vespignani A Velocityand hierarchical spread of epidemic outbreaks in scale-free networksPhys Rev Lett 92 178701 (2004)

53 Wang Y Chakrabarti D Wang G amp Faloutsos C in Proc 22ndInternational Symposium on Reliable Distributed Systems (SRDSrsquo03) 25ndash34(IEEE 2003)

54 Boguna M Pastor-Satorras R amp Vespignani A Absence of epidemicthreshold in scale-free networks with degree correlations Phys Rev Lett 90028701 (2003)

55 Castellano C amp Pastor-Satorras R Routes to thermodynamic limit onscale-free networks Phys Rev Lett 100 148701 (2008)

56 Chatterjee S amp Durrett R Contact processes on random graphs withpower law degree distributions have critical value 0 Ann Probab 372332ndash2356 (2009)

57 Castellano C amp Pastor-Satorras R Thresholds for epidemic spreading innetworks Phys Rev Lett 105 218701 (2010)

58 Durrett R Some features of the spread of epidemics and information on arandom graph Proc Natl Acad Sci USA 107 4491ndash4498 (2010)

59 Pastor-Satorras R amp Vespignani A Immunization of complex networksPhys Rev E 65 036104 (2001)

60 Cohen R Havlin S amp Ben-Avraham D Efficient immunization strategiesfor computer networks and populations Phys Rev Lett 91 247901 (2003)

61 Holme P Efficient local strategies for vaccination and network attackEurophys Lett 68 908ndash914 (2004)

62 Goldenberg J Shavitt Y Shir E amp Solomon S Distributive immunizationof networks against viruses using the lsquohoney-potrsquo architecture Nature Phys 1184ndash188 (2005)

63 Motter A E Zhou C S amp Kurths J Enhancing complex-networksynchronization Europhys Lett 69 334ndash340 (2005)

64 Motter A E Zhou C S amp Kurths J Network synchronization diffusionand the paradox of heterogeneity Phys Rev E 71 016116 (2005)

65 Goacutemez-Gardentildees J Campillo M Floria L M amp Moreno Y Dynamicalorganization of cooperation in complex topologies Phys Rev Lett 98108103 (2007)

66 Korniss G Synchronization in weighted uncorrelated complex networks in anoisy environment Optimization and connections with transport efficiencyPhys Rev E 75 051121 (2007)

67 Arenas A Diacuteaz-Guilera A amp Guimeragrave R Communication in networks withhierarchical branching Phys Rev Lett 86 3196ndash3199 (2001)

68 Guimeragrave R Arenas A Diacuteaz-Guilera A amp Giralt F Dynamical propertiesof model communication networks Phys Rev E 66 026704 (2002)

69 Sreenivasan S Cohen R Loacutepez E Toroczkai Z amp Stanley H EStructural bottlenecks for communication in networks Phys Rev E 75036105 (2007)

70 Castellano C Loreto V Barrat A Cecconi F amp Parisi D Comparisonof voter and Glauber ordering dynamics on networks Phys Rev E 71066107 (2005)

71 Sood V amp Redner S Voter model on heterogeneous graphs Phys Rev Lett94 178701 (2005)

72 Suchecki K Eguiacuteluz V M amp SanMiguel M Conservation laws for the votermodel in complex networks Europhys Lett 69 228ndash234 (2005)

73 Klemm K Eguiacuteluz V M Toral R amp San Miguel M Nonequilibriumtransitions in complex networks A model of social interaction Phys Rev E67 026120 (2003)

74 Santos F C Pacheco J M amp Lenaerts T Evolutionary dynamics of socialdilemmas in structured heterogeneous populations Proc Natl Acad Sci USA103 3490ndash3494 (2006)

75 van Kampen N G Stochastic Processes in Physics and Chemistry(North-Holland 1981)

76 Bolker B M amp Grenfell T Chaos and biological complexity in measlesdynamics Proc Trans R Soc Lond B 251 75ndash81 (1993)

77 Keeling M J amp Rohani P Estimating spatial coupling in epidemiologicalsystems A mechanistic approach Ecol Lett 5 20ndash29 (2002)

78 Sattenspiel L amp Dietz K A structured epidemic model incorporatinggeographic mobility among regionsMath Biosci 128 71ndash91 (1995)

79 Watts D Muhamad R Medina D C amp Dodds P S Multiscale resurgentepidemics in a hierarchical metapopulation model Proc Natl Acad Sci USA102 11157ndash11162 (2005)

80 Turing A M The chemical basis of morphogenesis Phil Trans R Soc LondB237 37ndash72 (1952)

81 Nakao H amp Mikhailov A S Turing patterns in network-organizedactivator-inhibitor systems Nature Phys 6 544ndash550 (2010)

82 Colizza V Pastor-Satorras R amp Vespignani A Reactionndashdiffusion processesand metapopulation models in heterogeneous networks Nature Phys 3276ndash282 (2007)

83 Colizza V amp Vespignani A Invasion threshold in heterogeneousmetapopulation networks Phys Rev Lett 99 148701 (2007)

84 Colizza V amp Vespignani A Epidemic modeling in metapopulation systemswith heterogeneous coupling pattern Theory and simulations J Theor Biol251 450ndash467 (2008)

85 Bartheacutelemy M Godregraveche C amp Luck J-M Fluctuation effects inmetapopulation models Percolation and pandemic threshold J Theor Biol267 554ndash564 (2010)

86 Saldana J Continuous-time formulation of reactionndashdiffusion processes onheterogeneous metapopulations Phys Rev E 78 012902 (2008)

87 Ni S amp Weng W Impact of travel patterns on epidemic dynamicsin heterogeneous spatial metapopulation networks Phys Rev E 79016111 (2009)

88 Ben-Zion Y Cohena Y amp Shnerba N M Modeling epidemics dynamics onheterogenous networks J Theor Biol 264 197ndash204 (2010)

89 Balcan D amp Vespignani A Phase transitions in contagion processes mediatedby recurrent mobility patterns Nature Phys 7 581ndash586 (2011)

90 Belik V Geisel T amp Brockmann D Natural human mobility patterns andspatial spread of infectious diseases Phys Rev X 1 011001 (2011)

91 Cooper B S Pitman R J Edmunds W J amp Gay N J Delaying theinternational spread of pandemic influenza PLoS Med 3 e12 (2006)

92 Hollingsworth T D Ferguson N M amp Anderson R M Will travelrestrictions control the international spread of pandemic influenza NatureMed 12 497ndash499 (2006)

93 Hufnagel L Brockmann D amp Geisel T Forecast and control of epidemicsin a globalized world Proc Natl Acad Sci USA 101 15124ndash15129 (2004)

94 Eubank S et al Modelling disease outbreaks in realistic urban social networksNature 429 180ndash184 (2004)

95 Longini I M et al Containing pandemic infleunza at the source Science 3091083ndash1087 (2005)

96 Ferguson N M et al Strategies for containing an emerging influenzapandemic in Southeast Asia Nature 437 209ndash211 (2005)

97 Colizza V Barrat A Barthlemy M Valleron M A J amp Vespignani AModeling the worldwide spread of pandemic influenza Baseline case andcontainment interventions PLoS Med 4 e13 (2007)

98 Balcan D et al Seasonal transmission potential and activity peaks of thenew influenza A(H1N1) A Monte Carlo likelihood analysis based on humanmobility BMCMed 7 45 (2009)

99 Merler S Ajelli M Pugliese A amp Ferguson N M Determinants of thespatiotemporal dynamics of the 2009H1N1 pandemic in Europe Implicationsfor real-time modelling PLoS Comput Biol 7 e1002205 (2011)

100 Gladwell M The Tipping Point How Little Things Can Make a Big Difference(Little Brown and Company 2002)

101 Helbing D amp Yu W The outbreak of cooperation among success-drivenindividuals under noisy condition Proc Natl Acad Sci USA 1063680ndash3685 (2009)

102 Xie J et al Social consensus through the influence of commited minoritiesPhys Rev E 84 011130 (2011)

103 Morris M amp Kretzschmar M Concurrent partnerships and the spread ofHIV AIDS 11 641ndash648 (1997)

104 Moody J The importance of relationship timing for diffusion Indirectconnectivity and STD infection risk Soc Forces 81 25ndash56 (2002)

105 Isella L et al Whatrsquos in a crowd Analysis of face-to-face behavioral networksJ Theor Biol 271 166ndash180 (2011)

106 Volz E amp Meyers L A Epidemic thresholds in dynamic contact networksJ R Soc Interface 6 233ndash241 (2009)

107 Holme P amp Newman M E J Nonequilibrium phase transition in thecoevolution of networks and opinions Phys Rev E 74 056108 (2006)

108 Centola D Gonzalez-Avella J C Eguiluz V M amp San Miguel MHomophily cultural drift and the co-evolution of cultural groups J ConflictResolution 51 905ndash929 (2007)

109 Funk S Salatheacute M amp Jansen V A A Modelling the inuence of humanbehaviour on the spread of infectious diseases A review J R Soc Interface 71247ndash1256 (2010)

110 Perra N Balcan D Goncalves B amp Vespignani A Towards acharacterization of behaviorndashdisease models PLoS ONE 6 e23084 (2011)

111 Bauch C T amp Earn D J Vaccination and the theory of games Proc NatlAcad Sci USA 101 13391ndash13394 (2004)

112 Liu Y-Y Slotine J-J amp Barabasi A-L Controllability of complex networksNature 473 167ndash173 (2011)

113 Conover M et al Proc 5th International Conference on Weblogs and SocialMedia (ICWSM) 89ndash96 (2011)

114 Ratkiewicz J et al Proc 20th International Conference Companion on WorldWide Web (WWW rsquo11) 249ndash252 (ACM 2001)

115 Kim B J Yoon C N Han S K amp Jeong H Path finding strategies inscale-free networks Phys Rev E 65 027103 (2002)

116 Adamic L A Lukose R M Puniyani A R amp Huberman B A Search inpower-law networks Phys Rev E 64 046135 (2001)

117 Brin S amp Page L The anatomy of a large-scale hypertextual Web searchengine Comput Netw ISDN Syst 30 107ndash117 (1998)

118 Bajardi P et al Human mobility networks travel restrictions and the globalspread of 2009 H1N1 pandemic PLoS ONE 6 e16591 (2011)

AcknowledgementsI thank B Goncalves and N Perra for their help with the figures and a critical reading ofthe manuscript This work has been partially funded by the NIH R21-DA024259DTRA-1-0910039 and NSF CCF-1101743 and NSF CMMI-1125095 awards The workhas been also partly sponsored by the Army Research Laboratory and was accomplishedunder Cooperative Agreement Number W911NF-09-2-0053 The views and conclusionscontained in this document are those of the authors and should not be interpreted asrepresenting the official policies either expressed or implied of the Army ResearchLaboratory or the US Government

Additional informationThe authors declare no competing financial interests Reprints and permissionsinformation is available online at httpwwwnaturecomreprints

PROGRESS ARTICLE | INSIGHTPUBLISHED ONLINE 22 DECEMBER 2011 | DOI 101038NPHYS2180

Networks formed from interdependent networksJianxi Gao12 Sergey V Buldyrev3 H Eugene Stanley1 and Shlomo Havlin4

Complex networks appear in almost every aspect of science and technology Although most results in the field have beenobtained by analysing isolated networks many real-world networks do in fact interact with and depend on other networks Theset of extensive results for the limiting case of non-interacting networks holds only to the extent that ignoring the presenceof other networks can be justified Recently an analytical framework for studying the percolation properties of interactingnetworks has been developed Here we review this framework and the results obtained so far for connectivity properties oflsquonetworks of networksrsquo formed by interdependent random networks

The interdisciplinary field of network science has attracted agreat deal of attention in recent years1ndash30 This development isbased on the enormous number of data that are now routinely

being collected modelled and analysed concerning social31ndash39economic14364041 technological4042ndash48 and biological9134950 sys-tems The investigation and growing understanding of this extraor-dinary volume of data will enable us to make the infrastructures weuse in everyday life more efficient andmore robust

The original model of networks random graph theory wasdeveloped in the 1960s by Erdős and Reacutenyi and is based on theassumption that every pair of nodes is randomly connected withthe same probability leading to a Poisson degree distribution Inparallel in physics lattice networks where each node has exactly thesame number of links have been studied tomodel physical systemsAlthough graph theory is a well-established tool in the mathematicsand computer science literature it cannot describe well modernreal-life networks Indeed the pioneering 1999 observation byBarabasi2 that many real networks do not follow the ErdősndashReacutenyimodel but that organizational principles naturally arise in mostsystems led to an overwhelming accumulation of supporting datanew models and computational and analytical results and to theemergence of a new science that of complex networks

Complex networks are usually non-homogeneous structuresthat in many cases obey a power-law form in their degree (thatis number of links per node) distribution These systems arecalled scale-free networks Real networks that can be approximatedas scale-free networks include the Internet3 the World WideWeb4 social networks31ndash39 representing the relations betweenindividuals infrastructure networks such as those of airlines51networks in biology9134950 in particular networks of proteinndashprotein interactions10 gene regulation and biochemical pathwaysand networks in physics such as polymer networks or the potential-energy-landscape network The discovery of scale-free networks ledto a re-evaluation of the basic properties of networks such as theirrobustness which exhibit a drastically different character than thoseof ErdősndashReacutenyi networks For example whereas homogeneousErdősndashReacutenyi networks are extremely vulnerable to random failuresheterogeneous scale-free networks are remarkably robust45 A greatpart of our current knowledge on networks is based on ideasborrowed from statistical physics such as percolation theoryfractals and scaling analysis An important property of theseinfrastructures is their stability and it is thus important that weunderstand and quantify their robustness in terms of node and

1Center for Polymer Studies and Department of Physics Boston University Boston Massachusetts 02215 USA 2Department of Automation ShanghaiJiao Tong University 800 Dongchuan Road Shanghai 200240 China 3Department of Physics Yeshiva University New York New York 10033 USA4Department of Physics Bar-Ilan University 52900 Ramat-Gan Israel e-mail havlinophirphbiuacil

link failures Percolation theory was introduced to study networkstability and predicted the critical percolation threshold5 Therobustness of a network is usually either characterized by the valueof the critical threshold analysed using percolation theory52 ordefined as the integrated size of the largest connected cluster duringthe entire attack process53 The percolation approach was alsoproved to be extremely useful in addressing other scenarios such asefficient attacks or immunization675455 and for obtaining optimalpaths56 aswell as for designing robust networks53 Network conceptshave also proven to be useful for the analysis and understanding ofthe spread of epidemics5758 and the organizational laws of socialinteractions such as friendships5960 or scientific collaborations6162Ref 63 investigated topologically biased failure in scale-freenetworks network and control of the robustness or fragility throughfine-tuning of the topological bias in the failure process

A large number of new measures and methods have beendeveloped to characterize network properties including measuresof node clustering network modularity correlation betweendegrees of neighbouring nodes measures of node importanceand methods for the identification and extraction of communitystructures These measures demonstrated that many real networksand in particular biological networks contain network motifsmdashsmall specific subnetworksmdashthat occur repeatedly and provideinformation about functionality9 Dynamical processes suchas flow and electrical transport in heterogeneous networkswere shown to be significantly more efficient when comparedwith ErdősndashReacutenyi networks6465 Furthermore it was shown thatnetworks can also possess self-similar properties so that underproper coarse graining (or renormalization) of the nodes thenetwork properties remain invariant19

However these complex systems were mainly modelled andanalysed as single networks that do not interact with or dependon other networks In interacting networks the failure of nodesin one network generally leads to the failure of dependentnodes in other networks which in turn may cause furtherdamage to the first network leading to cascading failures andcatastrophic consequences It is known for example that blackoutsin various countries have been the result of cascading failuresbetween interdependent systems such as communication andpower grid systems6768 Furthermore different kinds of criticalinfrastructure are also coupled together such as systems of waterand food supply communications fuel financial transactionsand power generation and transmission Modern technology has

NATURE PHYSICS DOI101038NPHYS2180 INSIGHT | PROGRESS ARTICLE

pc pc p

Pinfin

First order

Second order

Figure 1 | Schematic demonstration of first- and second-order percolationtransitions In the second-order case the giant component is continuouslyapproaching zero at the percolation threshold p= pc In the first-order casethe giant component approaches zero discontinuously

produced infrastructures that are becoming more and moreinterdependent and understanding how robustness is affected bythese interdependences is one of the main challenges faced whendesigning resilient infrastructures6769ndash72 In recent decades researchwas carried out in applied science on cataloguing analysing andmodelling the interdependences in critical infrastructure as wellas modelling cascading failures in coupled critical infrastructurenetworks4042ndash48 However no systematic mathematical frameworksuch as percolation theory is currently available for adequatelyaddressing the consequences of disruptions and failures occurringsimultaneously in interdependent critical infrastructures

Recently motivated by the fact that modern crucially importantinfrastructures significantly interact a mathematical frameworkwas developed73 to study percolation in a system of two inter-dependent networks subject to cascading failure The analyticalframework is based on a generating-function formalismwidely usedfor studies of percolation and structure within a single network73ndash75The framework for interdependent networks enables us to followthe dynamics of the cascading failures as well as to derive theanalytic solutions for the final steady state It was found73 thatcertain types of interdependent network were significantly morevulnerable than their non-interacting counterparts The failure ofeven a small number of elements within a single network maytrigger a catastrophic cascade of events that destroys the globalconnectivity For a fully interdependent case in which each nodein one network depends on a functioning node in other networksand vice versa a first-order discontinuous phase transition whichis dramatically different from the second-order continuous phasetransition found in isolated networks (Fig 1) was found73 Thisphenomenon is caused by the presence of two types of linkconnectivity links within each network and dependence linksbetween networks Connectivity links enable the network to carryout its function and dependence links represent the fact that thefunction of a given node in one network depends crucially onnodes in other networks The case of connectivity links betweenthe different networks was studied in ref 66 It was shown76

that when the dependence coupling between the networks isreduced at a critical coupling strength the percolation transitionbecomes second order

More recently two important generalizations of the basic modelof ref 73 have been developed

One generalization takes into account that in real-worldscenarios the initial failure of important nodes (or hubs) maybe not random but targeted A mathematical framework forunderstanding the robustness of interdependent networks underan initial targeted attack has been studied in ref 77 Theauthors of that work developed a general technique that uses therandom-attack problem to map the targeted-attack problem ininterdependent networks

The other generalization takes into account that in real-worldscenarios the assumption that each node in network A dependson one and only one node in network B and vice versa may notbe valid To correct this shortcoming a theoretical framework forunderstanding the robustness of interdependent networks with arandom number of support and dependence relationships has beendeveloped and studied78

In all of the above studies7376ndash78 the dependent pairs ofnodes in both networks were chosen randomly Thus when high-degree nodes in one network depend with a high probabilityon low-degree nodes of another network the configurationbecomes vulnerable To quantify and better understand thisphenomenon we proposed two lsquointersimilarityrsquo measures betweenthe interdependent networks79 On the one hand intersimilarityoccurs in interdependent networks when nodes with similar degreestend to be interdependent On the other hand it occurs if theneighbours of interdependent nodes in each network also tend tobe interdependent Refs 79ndash81 found that as the interdependentnetworks become more intersimilar the system becomes morerobust A system composed of an interdependent world-wideseaport and airport networks and the world-wide airport networkwas studied in ref 79 where it was found that well-connectedseaports tend to couple with well-connected airports and twoways of measuring the intersimilarity of interdependent networkswere developed The case in which all pairs of interdependentnodes in both networks have the same degree was solvedanalytically in ref 82

The robustness of a two-coupled-networks system has beenstudied for dependence coupling73 and for connectivity coupling66Very recently a more realistic coupled network system with bothdependence and connectivity links between the coupled networkswas studied83 Using a percolation approach rich andunusual phasetransition phenomena were found including a mixed first-orderand second-order hybrid transition This hybrid transition showsthat a discontinuous jump in the size of the giant component (as ina first-order transition) is followed by a continuous decrease to zero(as in a second-order transition)

Previous studies of isolated networks in which dependence linkscause cascading failure fall into two categories

The first studies failures due to network overload when thenetwork flow is a physical quantity for example in power trans-mission systems transportation networks or Internet traffic84ndash87The models produced by these studies demonstrate that when anoverloaded node stops traffic flow the choosing of alternative pathscan overload other nodes and a cascading failure that disables theentire network can result

The second is studies that produce models based on local depen-dences such as the decision-making of interacting agents11 In thesemodels the state of a node depends on the state of its neighboursthat is a failing node will cause its neighbours to also fail

The rich phenomena found in interdependent networks andthe insights obtained from the percolation framework developedin refs 7376 have led to a better understanding of the effect ofdependence links within single isolated networks A percolationapproach for a single network in the presence of randomdependence links was developed recently88ndash90 The results show thatcascading failures occur yielding a first-order transition and that

PROGRESS ARTICLE | INSIGHT NATURE PHYSICS DOI101038NPHYS2180

the percolation threshold of the network significantly increases withan increase in the number of dependence links

Generating functions for a single networkWe begin by describing the generating-function formalism74 for asingle network that will also be useful in studying interdependentnetworks We assume that all Ni nodes in network i are randomlyassigned a degree k from a probability distribution Pi(k) and arerandomly connected with the only constraint that the node withdegree k has exactly k links91 We define the generating function ofthe degree distribution

Gi(x)equivinfinsumk=0

Pi(k)xk (1)

where x is an arbitrary complex variable Using equation (1) theaverage degree of network i is

〈k〉i=infinsumk=0

kPi(k)=partGi

∣∣∣∣xrarr1=Gprimei(1) (2)

In the limit of infinitely large networks Nirarrinfin the randomconnection process can bemodelled as a branching process inwhichan outgoing link of any node has a probability kPi(k)〈k〉i of beingconnected to a node with degree k which in turn has kminus1 outgoinglinks Using equations (1) and (2) the generating function of thisbranching process is defined as

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Gprimei(x)Gprimei(1)

Let fi be the probability that a randomly selected link doesnot lead to the giant component If a link leads to a node withk minus 1 outgoing links this probability is f kminus1i Thus Hi(fi) alsohas the meaning that a randomly selected link does not lead tothe giant component and hence fi satisfies the recursive relationequation fi = Hi(fi) The probability that a node with degree kdoes not belong to the giant component is f ki and hence theprobability that a randomly selected node belongs to the giantcomponent is gi= 1minusGi(fi)

Once a fraction 1minus p of nodes is randomly removed from anetwork the generating function remains the same but with anew argument zi equiv px + 1minus p (ref 75) Accordingly owing tothe definition of fi and gi the probability that a randomly chosensurviving node belongs to a giant component is given by

gi(p)= 1minusGi[pfi(p)+1minusp] (4)

where fi(p) satisfies

fi(p)=Hi[pfi(p)+1minusp] (5)

Thus Pinfini the fraction of nodes that belongs to the giantcomponent is given by the product75

Pinfini= pgi(p) (6)

As p decreases the non-trivial solution fi lt 1 of equation (5)gradually approaches the trivial solution fi = 1 Accordingly Pinfinigradually approaches zero as in a second-order phase transition andbecomes zero when two solutions of equation (5) coincide at p=pcAt this point the straight line corresponding to the left-hand side

of equation (5) becomes tangent to the curve corresponding to itsright-hand side yielding

pc= 1H prime i(1) (7)

For example for ErdősndashReacutenyi networks92ndash94 characterized bya Poisson degree distribution using equations (1) (3) and (7)we obtain

Gi(x)=Hi(x)= exp[〈k〉i(xminus1)] (8)

gi(p)= 1minus fi(p) (9)

fi(p)= expp〈k〉i[fi(p)minus1] (10)

and using equations (7) and (8)

pc=1〈k〉i

Finally using equations (6) (9) and (10) we obtain a directequation for Pinfini

Pinfini= p[1minusexp(minus〈k〉iPinfini)] (12)

Framework of two partially interdependent networksA generalization of the percolation theory of two fully interdepen-dent networks73 has been developed by Parshani et al76 where amore realistic case of a pair of partially interdependent networkshas been studied In this case both interacting networks have acertain fraction of completely autonomous nodes whose functiondoes not directly depend on the nodes of the other network It hasbeen found that once the fraction of autonomous nodes increasesabove a certain threshold the abrupt collapse of the interdependentnetworks characterized by a first-order transition observed in ref 73changes at a critical coupling strength to a continuous second-order transition as in classical percolation theory52

In the following we describe in more detail the frameworkdeveloped in ref 76 This framework consists of two networks Aand B with the numbers of nodes NA and NB respectively Withinnetwork A the nodes are randomly connected by A edges withdegree distribution PA(k) whereas the nodes in network B arerandomly connected by B edges with degree distribution PB(k) Theaverage degrees of the networks A and B are a and b respectively Inaddition a fraction qA of network A nodes depends on the nodes innetwork B and a fraction qB of network B nodes depends on thenodes in network A We assume that a node from one networkdepends on no more than one node from the other networkand if node Ai depends on node Bj and Bj depends on Ak thenk = i The latter condition which we call a no-feedback condition(Fig 2) excludes configurations that completely collapse even forfully interdependent networks once a single node is removed78We assume that the initial removal of nodes from network Ais a fraction 1 minus p

Next we present the formalism for the cascade processstep by step (Fig 3) After an initial removal of nodes theremaining fraction of nodes in network A is ψ prime1 equiv p The initialremoval of nodes will disconnect some nodes from the giantcomponent The remaining functional part of network A thereforeconstitutes a fraction ψ1 =ψ

1gA(ψprime

1) of the network nodes wheregA(ψ prime1) is defined by equations (4) and (5) As a fraction qB ofnodes from network B depends on nodes from network A thenumber of nodes in network B that become non-functional is

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

Figure 2 | Differences between the feedback condition and no-feedbackcondition ab In the case of feedback (a) node A3 depends on node B2and node B3 6= B2 depends on node A3 whereas if there is no feedback (b)this is forbidden The blue links between the two networks show thedependence links and the red links in each network show the connectivitylinks which enable each network to function

(1minusψ1)qB= qB[1minusψ prime1gA(ψprime

1)] Accordingly the remaining fractionof network B nodes is φprime1=1minusqB[1minusψ prime1gA(ψ

1)] and the fraction ofnodes in the giant component of network B isφ1=φprime1gB(φ

1)Following this approach we can construct the sequence ψ primet and

φprimet of the remaining fraction of nodes at each stage of the cascade offailures The general form is given by

ψ prime1equiv p

φprime1= 1minusqB[1minuspgA(ψ prime1)]

ψ primet = p[1minusqA(1minusgB(φprimetminus1))]

φprimet = 1minusqB[1minuspgA(ψ primetminus1)]

To determine the state of the system at the end of the cascadeprocess we look atψ primeτ and φ

τ at the limit of τrarrinfin This limit mustsatisfy the equationsψ primeτ =ψ

τ+1 andφprime

τ =φprime

τ+1 because eventually theclusters stop fragmenting and the fractions of randomly removednodes at steps τ and τ +1 are equal Denoting ψ primeτ = x and φprimeτ = y we arrive in the stationary state at a system of two equationswith two unknowns

x = p1minusqA[1minusgB(y)]

y = 1minusqB[1minusgA(x)p](14)

The giant components of networks A and B at the end of thecascade of failures are respectively PinfinA = ψinfin = xgA(x) andPinfinB=φinfin= ygB(y) Figure 4 shows the excellent agreement forthe cascading failures in the giant component between computersimulations and the analytical results The analytical results wereobtained by recursive relations (13) where gA(ψ primet ) and gB(φprimet ) arecomputed using equations (9) and (10)

Equation (14) can be illustrated graphically by two curves cross-ing in the (xy) plane For sufficiently large qA and qB the curvesintersect at two points (0lt x00lt y0) and (x0lt x1lt 1y0lt y1lt 1)Only the second solution (x1y1) has a physical meaning As pdecreases the two solutions become closer to each other remaininginside the unit square (0lt xlt10lt ylt1) and at a certain thresh-old p= pc they coincide 0lt x0 = x1 = xc lt 1 0lt y0 = y1 = yc lt 1

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

Figure 3 | Description of the dynamic process of cascading failures on twopartially interdependent networks which can be generalized to n partiallyinterdependent networks The black nodes represent the survival nodesthe yellow node represents the initially attacked node the red nodesrepresent the nodes removed because they do not belong to the largestcluster and the blue nodes represent the nodes removed because theydepend on the failed nodes in the other network In each stage for onenetwork we first remove the nodes that depend on the failed nodes in theother network or on the initially attacked nodes Next we remove the nodesthat do not belong to the largest cluster of the network

For pltpc the non-trivial solution corresponding to the intersectionabruptly disappears Thus for sufficiently large qA and qB PinfinAand PinfinB as a function of p show a first-order phase transition AsqB decreases the intersection of the curves moves out of the unitsquare therefore for small enough qB PinfinA as a function of p showsa second-order phase transition For the graphical representation ofequation (14) and all possible solutions see Fig 3 in ref 76

In a recent study95 it was shown that a pair of interdependentnetworks can be designed to be more robust by choosing theautonomous nodes to be high-degree nodes This choice mitigatesthe probability of catastrophic cascading failure

Framework for a network of interdependent networksIn many real systems there are more than two interdependentnetworks and diverse infrastructuresmdashwater and food supplynetworks communication networks fuel networks financialtransaction networks or power-station networksmdashcan be coupledtogether6970 Understanding the way system robustness is affectedby such interdependences is one of the main challenges whendesigning resilient infrastructures

Here we review the generalization of the theory of a pairof interdependent networks7376 to a system of n interactingnetworks96 which can be graphically represented (Fig 5) as anetwork of networks (NON) We develop an exact analytical

10 20 30 40 50t

Theory

Simulation

Figure 4 | Cascade of failures in two partially interdependent ErdosndashReacutenyinetworks The giant component φt for every iteration of the cascadingfailures is shown for the case of a first-order phase transition with the initialparameters p=08505 a= b= 25 qA=07 and qB=08 In thesimulations N= 2times 105 with over 20 realizations The grey lines representdifferent realizations The squares represent the average over allrealizations and the black line is obtained from equation (13)

approach for percolation of an NON system composed of n fullyor partially interdependent randomly connected networks Theapproach is based on analysing the dynamical process of thecascading failures The results generalize the known results forpercolation of a single network (n= 1) and the n= 2 result foundin refs 7376 and show that whereas for n= 1 the percolationtransition is a second-order transition for ngt 1 cascading failuresoccur and the transition becomes first order Our results forn interdependent networks suggest that the classical percolationtheory extensively studied in physics and mathematics is a limitingcase of n = 1 of a general theory of percolation in NON As weshall discuss here this general theory has many features that are notpresent in the classical percolation theory

In our generalization each node in the NON is a network itselfand each link represents a fully or partially dependent pair ofnetworks We assume that each network i (i = 12 n) of theNON consists of Ni nodes linked together by connectivity linksTwo networks i and j form a partially dependent pair if a certainfraction qji gt 0 of nodes of network i directly depends on nodes ofnetwork j that is they cannot function if the nodes in network j onwhich they depend do not function Dependent pairs are connectedby unidirectional dependence links pointing from network j tonetwork i This convention symbolizes the fact that nodes innetwork i receive supply from nodes in network j of a crucialcommodity for example electric power if network j is a power grid

We assume that after an attack or failure only a fraction of nodespi in each network i will remain We also assume that only nodesthat belong to a giant connected component of each network iwill remain functional This assumption helps explain the cascadeof failures nodes in network i that do not belong to its giantcomponent fail causing failures of nodes in other networks thatdepend on the failing nodes of network i The failure of these nodescauses the direct failure of the dependent nodes in other networksfailures of isolated nodes in them and further failure of nodes innetwork i and so on Our goal is to find the fraction of nodes Pinfiniof each network that remain functional at the end of the cascadeof failures as a function of all fractions pi and all fractions qij We assume that all networks in the NON are randomly connectednetworks characterized by a degree distribution of linksPi(k) wherek is a degree of a node in network i We further assume that each

qikqki

qi3qi2

Figure 5 | Schematic representation of a NON Circles representinterdependent networks and the arrows connect the partiallyinterdependent pairs For example a fraction of q3i of nodes in network idepend on the nodes in network 3 The networks that are not connected bythe dependence links do not have nodes that directly depend onone another

node a in network i may depend with probability qji on only onenode b in network j

We can study different models of cascading failures in whichwe vary the survival time of the dependent nodes after the failureof the nodes in other networks on which they depend and thesurvival time of the disconnected nodes We conclude that thefinal state of the networks does not depend on these details butcan be described by a system of equations somewhat analogousto the Kirchhoff equations for a resistor network This systemof equations has n unknowns xi These represent the fractionsof nodes that survive in network i after the nodes that fail inthe initial attack are removed and also the nodes dependingon the failed nodes in other networks at the end of cascadingfailure are removed but without considering yet the furtherfailing of nodes due to the internal connectivity of the networkThe final giant component of each network can be found fromthe equation Pinfini = xigi(xi) where gi(xi) is the fraction of theremaining nodes of network i that belong to its giant componentgiven by equation (4)

First we shall discuss the more complex case of the no-feedbackcondition The unknowns xi satisfy the systemof n equations

xi= piKprodj=1

[qjiyjigj(xj)minusqji+1] (15)

where the product is taken over the K networks interlinked withnetwork i by the partial dependence links (Fig 3) and

yij =xi

qjiyjigj(xj)minusqji+1(16)

has the meaning of the fraction of nodes in network j that surviveafter the damage from all the networks connected to networkj except network i is taken into account The damage fromnetwork imust be excluded owing to the no-feedback condition Inthe absence of the no-feedback condition equation (15) becomesmuch simpler as yji = xj Equation (15) is valid for any caseof interdependent NON whereas equation (16) represents theno-feedback condition

Four examples of a NON solvable analyticallyIn this section we present four examples that can be explicitlysolved analytically (1) a tree-like ErdősndashReacutenyi fully dependent

Chain-like NON Star-like NON Tree-like NON

Figure 6 | Three types of loopless NON composed of five couplednetworks All have the same percolation threshold and the same giantcomponent The dark node represents the origin network on which failuresinitially occur

NON (2) a tree-like random regular fully dependent NON (3) aloop-like ErdősndashReacutenyi partially dependent NON and (4) a randomregular network of partially dependent ErdősndashReacutenyi networksAll cases represent different generalizations of percolation theoryfor a single network In all examples except (3) we apply theno-feedback condition

(1) We solve explicitly96 the case of a tree-like NON (Fig 6)formed by n ErdősndashReacutenyi networks92ndash94 with the same averagedegrees k p1= p pi= 1 for i 6= 1 and qij = 1 (fully interdependent)From equations (15) and (16) we obtain an exact expression for theorder parameter the size of the mutual giant component for all p kand n values

Pinfin= p[1minusexp(minuskPinfin)]n (17)

Equation (17) generalizes known results for n= 12 For n= 1 weobtain the known result pc=1k equation (11) of an ErdősndashReacutenyinetwork and Pinfin(pc) = 0 which corresponds to a continuoussecond-order phase transition Substituting n= 2 in equation (17)yields the exact results of ref 73

Solutions of equation (17) are shown in Fig 7a for several valuesof n The special case n= 1 is the known ErdősndashReacutenyi second-orderpercolation law equation (12) for a single network In contrastfor any ngt 1 the solution of (17) yields a first-order percolationtransition that is a discontinuity of Pinfin at pc

Our results show (Fig 7a) that the NON becomes more vul-nerable with increasing n or decreasing k (pc increases whenn increases or k decreases) Furthermore for a fixed n whenk is smaller than a critical number kmin(n) pc ge 1 meaningthat for k lt kmin(n) the NON will collapse even if a singlenode fails96

(2) In the case of a tree-like network of interdependent randomregular networks97 where the degree k of each node in each networkis assumed to be the same we obtain an exact expression for theorder parameter the size of the mutual giant component for allp k and n values

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Numerical solutions of equation (18) are in excellent agreementwith simulations Comparing with the results of the tree-likeErdősndashReacutenyi NON we find that the robustness of n interdependentrandom regular networks of degree k is significantly higher thanthat of the n interdependent ErdősndashReacutenyi networks of averagedegree k Moreover whereas for an ErdősndashReacutenyi NON there existsa critical minimum average degree k = kmin that increases with n(below which the system collapses) there is no such analogous kminfor the random regular NON system For any k gt 2 the randomregular NON is stable that is pc lt 1 In general this is correctfor any network with any degree distribution Pi(k) such that

Pi(0)= Pi(1)= 0 that is for a network without disconnected orsingly connected nodes97

(3) In the case of a loop-like NON (for dependences inone direction) of n ErdősndashReacutenyi networks96 all the links areunidirectional and the no-feedback condition is irrelevant If theinitial attack on each network is the same 1minusp qiminus1i= qn1= q andki=k using equations (15) and (16)we obtain thatPinfin satisfies

Pinfin= p(1minuseminuskPinfin)(qPinfinminusq+1) (19)

Note that if q = 1 equation (19) has only a trivial solutionPinfin = 0 whereas for q = 0 it yields the known giant componentof a single network equation (12) as expected We presentnumerical solutions of equation (19) for two values of q inFig 7b Interestingly whereas for q = 1 and tree-like structuresequations (17) and (18) depend on n for loop-like NON structuresequation (19) is independent of n

(4) For NONs where each ER network is dependent on exactlym other ErdősndashReacutenyi networks (the case of a random regularnetwork of ErdősndashReacutenyi networks) we assume that the initial attackon each network is 1minus p and each partially dependent pair hasthe same q in both directions The n equations of equation (15)are exactly the same owing to symmetries and hence Pinfin can beobtained analytically

Pinfin=p2m

(1minuseminuskPinfin)[1minusq+radic(1minusq)2+4qPinfin]m (20)

from which we obtain

k(1minusq)m(21)

Again as in case (3) it is surprising that both the critical thresholdand the giant component are independent of the number ofnetworks n in contrast to tree-like NON (equations (17) and (18))but depend on the coupling q and on both degrees k andm Numerical solutions of equation (20) are shown in Fig 7cand the critical thresholds pc in Fig 7c coincide with thetheory equation (21)

Remark on scale-free networksThe above examples regarding ErdősndashReacutenyi and random regularnetworks have been selected because they can be explicitlysolved analytically In principle the generating function formalismpresented here can be applied to randomly connected networkswith any degree distribution The analysis of the scale-free networkswith a power-law degree distribution P(k) sim kminusλ is extremelyimportant because many real networks can be approximatedby a power-law degree distribution such as the Internet theairline network and social-contact networks such as networksof scientific collaboration21051 Analysis of fully interdependentscale-free networks73 shows that for interdependent scale-freenetworks pc gt 0 even in the case λ le 3 for which in a singlenetwork pc = 0 In general for fully interdependent networksthe broader the degree distribution the greater pc for networkswith the same average degree73 This means that networks with abroad degree distribution become less robust than networks witha narrow degree distribution This trend is the opposite of thetrend found in non-interacting isolated networks The explanationof this phenomenon is related to the fact that in randomlyinterdependent networks the hubs in one network may depend onpoorly connected nodes in another Thus the removal of a randomlyselected node in one network may cause a failure of a hub ina second network which in turn renders many singly connected

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

Figure 7 | The fraction of nodes in the giant component Pinfin as a function of p for three different examples a A tree-like fully (q= 1) interdependentNON Pinfin is shown as a function of p for k= 5 and several values of n The results are obtained using equation (17) Note that increasing n from n= 2 yieldsa first-order transition b A loop-like NON Pinfin is shown as a function of p for k=6 and two values of q The results are obtained using equation (19) Notethat increasing q yields a first-order transition c A random regular network of ErdosndashReacutenyi networks Pinfin is shown as a function of p for two different valuesof m when q=05 The results are obtained using equation (20) and the number of networks n can be any number with the condition that any network inthe NON connects exactly to m other networks Note that changing m from 2 to mgt 2 changes the transition from second order to first order (for q=05)

nodes non-functional and the multiplying damage travels backto the first network This explanation is corroborated by theanalytical proof in ref 82 which shows that if the degrees of theinterdependent nodes coincide then a network with a broaderdegree distribution will become more robust than a network witha narrower degree distribution that is the behaviour characteristicof non-interacting networks is restored Ref 82 also reports thatfor fully interdependent scale-free networks with equal degrees ofinterdependent pairs pc = 0 for λlt 3 Moreover the percolationtransition is a discontinuous first-order phase transition if and onlyif H primei (1)ltinfin that is if the degree distribution has a finite secondmoment For fully interdependent networks with uncorrelateddegrees of interdependent nodes the percolation transition isalways a discontinuous phase transition7376 These results as well asthe results of ref 79 show the need to studymore realistic situationsin which the interdependent networks have various correlationsin the dependences and connectivities A recent study of partiallyinterdependent scale-free networks shows that although the giantcomponent decreases significantly owing to cascading failures pc isalways zero as long as qlt1 (D Zhou et al unpublished)

Remaining challengesWe have reviewed recent studies of the robustness of a system ofinterdependent networks In interacting networks when a nodein one network fails it usually causes dependent nodes in othernetworks to fail which in turn may cause further damage in thefirst network and results in a cascade of failures with catastrophicconsequences Our analytical framework enables us to follow thedynamic process of the cascading failures step by step and toderive steady-state solutions Interdependent networks appear inall aspects of life nature and technology Transportation systemsinclude railway networks airline networks and other transportationsystems Some properties of interacting transportation systemshave been studied recently7980 In the field of physiology thehuman body can be regarded as a system of interdependentnetworks Examples of such interdependent NON systems includethe cardiovascular system the respiratory system the brain neuronsystem and the nervous system In biology the function of eachprotein is determined by its interacting proteins which can bedescribed by a network As many proteins are involved in anumber of different functions the protein-interaction system canbe regarded as a system of interacting networks In the field ofeconomics networks of banks insurance companies and businessfirms are interdependent

Thus far only a very few real-world interdependent systems havebeen analysed using the percolation approach717980 We expect ourpresent work to provide insights leading to a further analysis ofreal data on interdependent networks The benchmark models wepresent here can be used to study the structural functional androbustness properties of interdependent networks Because in realNONs individual networks are not randomly connected and theirinterdependent nodes are not selected at random it is crucial thatwe understand themany types of correlation that exist in real-worldsystems and that we further develop the theoretical tools to includesuch correlations Further studies of interdependent networksshould focus on an analysis of real data from many differentinterdependent systems and on the development of mathematicaltools for studying real-world interdependent systems

Many real-world networks are embedded in space and thespatial constraints strongly affect their properties30 We need tounderstand how these spatial constraints influence the robustnessproperties of interdependent networks7980 Other properties thatinfluence the robustness of single networks such as the dynamicnature of the configuration in which links or nodes appear anddisappear and the directed nature of some links as well as problemsassociated with degreendashdegree correlations and clustering shouldbe also addressed in future studies of coupled network systems It isalso important to investigate the case when a node in one networkis supplied by multiple nodes in an interdependent network Inrealistic interdependent pairs of networks i and j a node in networkimay depend on s supply nodes in network j and the total supply ofa commodity received by this node from network j must be greaterthan a certain threshold sc In the case of sc=0 and random selectionof the supply nodes this problem was solved in ref 78 for two in-terdependent networks and this solution can be straightforwardlygeneralized for an arbitraryNONby replacing equation (15)with

xi= piKprodj=1

1minusqjiGji[1minusxjgj(xj)] (22)

where Gji(x) is the generating function of the distribution of thesupply degree s of nodes in network i that depend on the supplyfrom nodes in network j When s= 1 for all such nodes Gji(x)= xand equation (22) reduces to equation (15) with yji = xj that is inthe absence of the no-feedback condition More complex cases ofmultiple supply nodes await further investigation

It is very important to find a way of improving the robustnessof interdependent infrastructures Our studies thus far show that

there are three methods to achieve this goal increase the fraction ofautonomous nodes76 particularly nodes with high degree95 designthe dependence links such that they connect the nodes with similardegrees7982 and protect the high-degree nodes against attack95

A coupled network in which the interlinks that is the linksbetween different networks are connectivity links was studied inref 66 The robustness of this system is greatly improved whencompared with a system in which the interlinks are dependencelinks A systematic study of the competing effects of aNON inwhichthe interlinks are both dependence and connectivity interlinks isneeded Interesting results on a model containing both dependenceand connectivity interlinks have been obtained83 Finally wemention an early study of the Ising model on coupled networks98Also interacting networks with respect to climate systems werestudied in ref 99

References1 Watts D J amp Strogatz S H Collective dynamics of lsquosmall-worldrsquo networks

Nature 393 440ndash442 (1998)2 Barabaacutesi A L amp Albert R Emergence of scaling in random networks Science

286 509ndash512 (1999)3 Faloutsos M Faloutsos P amp Faloutsos C On power-law relationships of the

internet topology Comput Commun Rev 29 378ndash382 (2000)4 Albert R Jeong H amp Barabaacutesi A L Error and attack tolerance of complex

networks Nature 406 378ndash382 (2000)5 Cohen R Erez K Ben-Avraham D amp Havlin S Resilience of the Internet to

random breakdown Phys Rev Lett 85 4626ndash4628 (2000)6 Callaway D S Newman M E J Strogatz S H amp Watts D J Network

robustness and fragility Percolation on random graphs Phys Rev Lett 855468ndash5471 (2000)

7 Cohen R Erez K Ben-Avraham D amp Havlin S Breakdown of the Internetunder intentional attack Phys Rev Lett 86 3682ndash3685 (2001)

8 Strogatz S H Exploring complex networks Nature 410 268ndash276 (2001)9 Milo R et al Network motifs Simple building blocks of complex networks

Science 298 824ndash827 (2002)10 Albert R amp Barabaacutesi A L Statistical mechanics of complex networks

Rev Mod Phys 74 47ndash97 (2002)11 Watts D J A simple model of global cascades on random networks Proc Natl

Acad Sci USA 99 5766ndash5771 (2002)12 Newman M E J The structure and function of complex networks SIAM Rev

45 167ndash256 (2003)13 Dorogovtsev S NampMendes J F FEvolution ofNetworks FromBiologicalNets

to the Internet and WWW (Physics) (Oxford Univ Press 2003)14 Bonanno G Caldarelli G Lillo F amp Mantegna R N Topology of

correlation-based minimal spanning trees in real and model markets PhysRev E 68 046130 (2003)

15 Barrat A Barthelemy M Pastor-Satorras R amp Vespignani A Thearchitecture of complex weighted networks Proc Natl Acad Sci USA 1013747ndash3752 (2004)

16 Newman M E J amp Girvan M Finding and evaluating community structurein networks Phys Rev E 69 026113 (2004)

17 Satorras R P amp Vespignani A Evolution and Structure of the Internet AStatistical Physics Approach (Cambridge Univ Press 2004)

18 Gallos L K Cohen R amp Argyrakis P et al Stability and topology ofscale-free networks under attack and defense strategies Phys Rev Lett 94188701 (2005)

19 Song C Havlin S ampMakse H A Self-similarity of complex networksNature433 392ndash395 (2005)

20 Boccaletti S Latora V Moreno Y Chavez M amp Hwang D U Complexnetworks Structure and dynamics Phys Rep 424 175ndash308 (2006)

21 Newman M E J Barabaacutesi A-L amp Watts D J The Structure and Dynamics ofNetworks (Princeton Univ Press 2006)

22 Caldarelli G amp Vespignani A Large Scale Structure and Dynamics of ComplexWebs (World Scientific 2007)

23 Barraacutet A Bartheacutelemy M amp Vespignani A Dynamical Processes on ComplexNetworks (Cambridge Univ Press 2008)

25 Kitsak M et al Identification of influential spreaders in complex networksNature Phys 6 888ndash893 (2010)

26 Newman M E J Networks An Introduction (Oxford Univ Press 2010)27 Cohen R amp Havlin S Complex Networks Structure Robustness and Function

(Cambridge Univ Press 2010)28 West B J amp Grigolini P Complex Webs Anticipating the Improbable

(Cambridge Univ Press 2011)

29 Bartheacutelemy M Spatial networks Phys Rep 499 1ndash101 (2011)30 Li D Kosmidis K Bunde A amp Havlin S Dimension of spatially embedded

networks Nature Phys 7 481ndash484 (2011)31 Snijders T A B Pattison P E Robins G L amp Handcock M S New

specifications for exponential random graph models Sociol Methodol 3699ndash153 (2006)

32 Borgatti S P Identifying sets of key players in a networkComput Math Org Theor 12 21ndash34 (2006)

34 Faust K amp Zvezki M Comparing social networks Size density and localstructure Linear Algebr Appl 3 185ndash216 (2006)

35 Handcock M S Raftery A E amp Tantrum J M Model-based clustering forsocial networks J R Stat Soc A 170 301ndash354 (2007)

36 Jackson M O amp Rogers B W Meeting strangers and friends of friends Howrandom are social networks Am Econom Rev 97 890ndash915 (2007)

37 Kleinberg J The convergence of social and technological networksCommun ACM 51 66ndash72 (2008)

38 Liben-Nowell D amp Kleinberg J Tracing information flow on a globalscale using internet chain-letter data Proc Natl Acad Sci USA 1054633ndash4638 (2008)

39 Borgatti S P Mehra A Brass D amp Labianca G Network analysis in thesocial sciences Science 323 892ndash895 (2009)

40 Joost R Inoperability inputndashoutput modeling of disruptions to interdependenteconomic systems Syst Eng 9 20ndash34 (2006)

41 Jackson M O Social and Economic Networks (Economics Physics Sociology)(Princeton Univ Press 2008)

42 Zimmerman R Decision-making and the vulnerability of interdependentcritical infrastructure 2004 IEEE Int Conf Syst Man Cybern 54059ndash4063 (2005)

43 Mendonca D amp Wallace W A Impacts of the 2001 World Trade Centerattack on New York City critical infrastructures J Infrast Syst 12260ndash270 (2006)

44 Robert B Morabito L amp Christie R D The operational tools formanaging physical interdependencies among critical infrastructuresInt J Crit Infrastruct 4 353ndash367 (2008)

45 Reed D A Kapur K C amp Christie R D Methodology for assessing theresilience of networked infrastructure IEEE Syst J 3 174ndash180 (2009)

46 Bagheri E amp Ghorbani A A UML-CI A reference model for profiling criticalinfrastructure systems Inform Syst Front 12 115ndash139 (2009)

47 Mansson D Thottappillil R Backstrom M amp Ludvika H V VMethodology for classifying facilities with respect to intentional EMIIEEE Trans Electromagn Compat 95 46ndash52 (2009)

48 Johansson J amp Hassel H An approach for modelling interdependentinfrastructures in the context of vulnerability analysis Reliab Eng Syst Saf 951335ndash1344 (2010)

49 Alon U Biological networks The tinkerer as an engineer Science 3011866ndash1867 (2003)

50 Khanin R amp Wit E How scale-free are biological networks J Comput Biol13 810ndash818 (2006)

51 Colizza V Barrat A Barthelemy M amp Vespignani A Prediction andpredictability of global epidemics The role of the airline transportationnetwork Proc Natl Acad Sci USA 103 2015ndash2020 (2006)

52 Bunde A amp Havlin S Fractals and Disordered Systems (Springer 1996)53 Schneider C M Arauacutejo N A M Moreira A A Havlin S amp Herrmann

H J Mitigation of malicious attacks on networks Proc Natl Acad Sci USA108 3838ndash3841 (2011)

54 Cohen R Havlin S amp Ben-Avraham D Efficient immunization strategies forcomputer networks and populations Phys Rev Lett 91 247901 (2003)

55 Chen Y Paul G Havlin S Liljeros F amp Stanley H E Finding a betterimmunization strategy Phys Rev Lett 101 058701 (2008)

56 Braunstein L A Buldyrev S V Cohen Havlin S amp Stanley H E Optimalpaths in disordered complex networks Phys Rev Lett 91 168701 (2003)

57 Pastor-Satorras R amp Vespignani A Epidemic spreading in scale-free networkPhys Rev Lett 86 3200ndash3203 (2001)

58 Balcan D et al Multiscale mobility networks and the large scale spreading ofinfectious diseases Proc Natl Acad Sci USA 106 21484ndash21489 (2009)

59 Palla G Derenyi I Farkas I amp Vicsek T Uncovering the overlappingcommunity structure of complex networks in nature and society Nature 435814ndash818 (2005)

60 Kossinets G amp Watts D Empirical analysis of an evolving social networkScience 311 88ndash90 (2006)

61 Newman M E J The structure of scientific collaboration networks Proc NatlAcad Sci USA 98 404ndash409 (2001)

63 Moreira A A Andrade J S Jr Herrmann H J amp Indekeu J O How tomakea fragile network robust and vice versa Phys Rev Lett 102 019701 (2009)

64 Lopez E Buldyrev S V Havlin S amp Stanley H E Anomalous transport inscale-free networks Phys Rev Lett 94 248701 (2005)

65 Boguntildeaacute M amp Krioukov D Navigating ultrasmall worlds in ultrashort timePhys Rev Lett 102 058701 (2009)

66 Leicht E A amp DrsquoSouza R M Percolation on interacting networks Preprint athttparxivorgabs09070894 (2009)

67 Rosato V Modeling interdependent infrastructures using interactingdynamical models Int J Crit Infrastruct 4 63ndash79 (2008)

68 USndashCanada Power System Outage Task Force Final Report on the August 14th2003 Blackout in the United States and Canada Causes and Recommendations(The Task Force 2004)

69 Peerenboom J Fischer R amp Whitfield R in Proc CRISDRMIIITNSFWorkshop Mitigating the Vulnerability of Critical Infrastructures to CatastrophicFailures (2001)

70 Rinaldi S Peerenboom J amp Kelly T Identifying understanding andanalyzing critical infrastructure interdepedencies IEEE Control Syst Magn 2111ndash25 (2001)

71 Yagan O Qian D Zhang J amp Cochran D Optimal allocation ofinterconnecting links in cyber-physical systems Interdependence cascadingfailures and robustness httpwwweceumdedusimoyaganJournalsInterdependent_Journalpdf (2011)

72 Vespignani A The fragility of interdependency Nature 464 984ndash985 (2010)73 Buldyrev S V Parshani R Paul G Stanley H E amp Havlin S

Catastrophic cascade of failures in interdependent networks Nature464 1025ndash1028 (2010)

74 Newman M E J Strogatz S H amp Watts D J Random graphs with arbitrarydegree distributions and their applications Phys Rev E 64 026118 (2001)

75 Shao J Buldyrev S V Braunstein L A Havlin S amp Stanley H E Structureof shells in complex networks Phys Rev E 80 036105 (2009)

76 Parshani R Buldyrev S V amp Havlin S Interdependent networks Reducingthe coupling strength leads to a change from a first to second order percolationtransition Phys Rev Lett 105 048701 (2010)

77 Huang X Gao J Buldyrev S V Havlin S amp Stanley H E Robustnessof interdependent networks under targeted attack Phys Rev E (R) 83065101 (2011)

78 Shao J Buldyrev S V Havlin S amp Stanley H E Cascade of failuresin coupled network systems with multiple support-dependence relationsPhys Rev E 83 036116 (2011)

79 Parshani R Rozenblat C Ietri D Ducruet C amp Havlin S Inter-similaritybetween coupled networks Europhys Lett 92 68002ndash68006 (2010)

80 Gu C et al Onset of cooperation between layered networks Phys Rev E 84026101 (2011)

81 Cho W Coh K amp Kim I Correlated couplings and robustness of couplednetworks Preprint at httparxivorgabs10104971 (2010)

82 Buldyrev S V Shere N W amp Cwilich G A Interdependent networks withidentical degrees of mutually dependent nodes Phys Rev E 83 016112 (2011)

83 Hu Y Ksherim B Cohen R amp Havlin S Percolation in interdependent andinterconnected networks Abrupt change from second to first order transitionPhys Rev E (in the press) Preprint at httparxivorgabs11064128 (2011)

84 Sachtjen M L Carreras B A amp Lynch V E Disturbances in a powertransmission system Phys Rev E 61 4877ndash4882 (2000)

85 Motter A E amp Lai Y C Cascade-based attacks on complex networksPhys Rev E 66 065102 (2002)

86 Moreno Y Pastor S R Vaacutezquez A amp Vespignani A Critical loadand congestion instabilities in scale-free networks Europhys Lett 62292ndash298 (2003)

87 Motter A E Cascade control and defense in complex networks Phys Rev Lett93 098701 (2004)

88 Parshani R Buldyrev S V amp Havlin S Critical effect of dependencygroups on the function of networks Proc Natl Acad Sci USA 1081007ndash1010 (2011)

89 Bashan A Parshani R amp Havlin S Percolation in networks composed ofconnectivity and dependency links Phys Rev E 83 051127 (2011)

90 Bashan A amp Havlin S The combined effect of connectivity and dependencylinks on percolation of networks J Stat Phys 145 686ndash695 (2011)

91 Molloy M amp Reed B The size of the giant component of a random graph witha given degree sequence Combin Probab Comput 7 295ndash305 (1998)

92 Erdős P amp Reacutenyi A On random graphs I Publ Math 6 290ndash297 (1959)93 Erdős P amp Reacutenyi A On the evolution of random graphs Inst Hung Acad Sci

5 17ndash61 (1960)94 Bollobaacutes B Random Graphs (Academic 1985)95 Schneider C M Arauacutejo N A M Havlin S amp Herrmann H J

Towards designing robust coupled networks Preprint at httparxivorgabs11063234 (2011)

96 Gao J Buldyrev S V Havlin S amp Stanley H E Robustness of a network ofnetworks Phys Rev Lett 107 195701 (2011)

97 Gao J Buldyrev S V Havlin S amp Stanley H E Robustness of a tree-likenetwork of interdependent networks Preprint athttparxivorgabs11085515 (2011)

98 Suchecki K amp Holyst J A Ising model on two connected BarabasindashAlbertnetworks Phys Rev E 74 011122 (2006)

99 Donges J F Schultz H C H Marwan N Zou Y amp Kurths J Investigatingthe topology of interacting networks Eur Phys J B (2011 in the press)

AcknowledgementsWe thank R Parshani for helpful discussions We thank the DTRA (Defense ThreatReduction Agency) and the Office of Naval Research for support JG also thanks theShanghai Key Basic Research Project (grant no 09JC1408000) and the National NaturalScience Foundation of China (grant no 61004088) for support SVB acknowledges thepartial support of this research through the B W Gamson Computational ScienceCenter at Yeshiva College SH thanks the European EPIWORK project DeutscheForschungsgemeinschaft (DFG) and the Israel Science Foundation for financial support

Additional informationThe authors declare no competing financial interests Reprints and permissionsinformation is available online at httpwwwnaturecomreprints Correspondence andrequests for materials should be addressed to HES

nphys2188pdf

The network takeover
Network universe A visualization of the first large-scale network explicitly mapped out to explore the large-scale structure of real networks The map was generated in 1999 and represents a small portion of the World Wide Web11 this map has led to the d
References
nphys2190pdf
Between order and chaos
Complicated yes but is it complex
Complexities
Applications
Closing remarks
Figure 1 ε-machines for four information sources
Figure 2 Structure versus randomness
Figure 3 Complexity--entropy diagrams
References
Acknowledgements
Additional information
nphys2162pdf
Communities modules and large-scale structure in networks
Hierarchical clustering
Optimization methods
Block models
Conclusion
Figure 1 Example network showing community structure
Figure 2 A network of collaborations among scientists at a research institute
Figure 3 Average-linkage clustering of a small social network
Figure 4 Analysis of a network of links between web sites about US politics
Figure 5 Hierarchical divisions in a food web of grassland species
References
Acknowledgements
nphys2160pdf
Modelling dynamical processes in complex socio-technical systems
Dynamical processes and phase transitions
Complex networks and dynamical processes
Reaction--diffusion processes and computational thinking
Co-evolution timescale and control
Conclusions
Figure 1 Phase diagram of epidemic models
Figure 2 Progression of an epidemic process
Figure 3 Illustration of the global threshold in reaction--diffusion processes
Figure 4 Visualization of the dynamical network generated by Twitter interactions
Box 1 The heterogeneous mean-field approach
Box 2 The particle--network framework
References
Acknowledgements
nphys2180pdf
Networks formed from interdependent networks
Generating functions for a single network
Framework of two partially interdependent networks
Framework for a network of interdependent networks
Four examples of a NON solvable analytically
Remark on scale-free networks
Remaining challenges
Figure 1 Schematic demonstration of first- and second-order percolation transitions
Figure 2 Differences between the feedback condition and no-feedback condition
Figure 3 Description of the dynamic process of cascading failures on two partially interdependent networks which can be generalized to n partially interdependent networks
Figure 4 Cascade of failures in two partially interdependent Erdős--Reacutenyi networks
Figure 5 Schematic representation of a NON
Figure 6 Three types of loopless NON composed of five coupled networks
Figure 7 The fraction of nodes in the giant component Pinfty as a function of p for three different examples
References
Acknowledgements

INSIGHT | CONTENTS

NPG LONDONThe Macmillan Building 4 Crinan Street London N1 9XWT +44 207 833 4000 F +44 207 843 4563naturephysicsnaturecom

EDITOR ALISON WRIGHT

INSIGHT EDITOR ANDREAS TRABESINGER

PRODUCTION EDITOR JENNY MARSDEN

COPY EDITOR MELANIE HANVEY

ART EDITOR KAREN MOORE

EDITORIAL ASSISTANTAMANDA WYATT

MARKETINGSARAH-JANE BALDOCK

PUBLISHERRUTH WILSON

EDITOR-IN-CHIEF NATURE PUBLICATIONSPHILIP CAMPBELL

COVER IMAGE In many large ensembles the property of the system as a whole cannot be understood by studying the individual entities mdash neurons in the brain for example or transport users in traffic networks The past decade however has seen important progress in our fundamental understanding of what such seemingly disparate lsquocomplex systemsrsquo have in common Image copy Marvin E NewmanGetty Images

A formal definition of what constitutes a complex system is not easy to devise equally difficult is the

delineation of which fields of study fall within the bounds of lsquocomplexityrsquo An appealing approach mdash but only one of several possibilities mdash is to play on the lsquomore is differentrsquo theme declaring that the properties of a complex system as a whole cannot be understood from the study of its individual constituents There are many examples from neurons in the brain to transport users in traffic networks to data packages in the Internet

Large datasets mdash collected for example in proteomic studies or captured in records of mobile-phone users and Internet traffic mdash now provide an unprecedented level of information about these systems Indeed the availability of these detailed datasets has led to an explosion of activity in the modelling of complex systems Data-based models can not only provide an understanding of the properties and behaviours of individual systems but also beyond that might lead to the discovery of common properties between seemingly disparate systems

Much of the progress made during the past decade or so comes under the banner of lsquonetwork sciencersquo The representation of complex systems as networks or graphs

has proved to be a tremendously useful abstraction and has led to an understanding of how many real-world systems are structured what kinds of dynamic processes they support and how they interact with each other This Nature Physics Insight is therefore admittedly inclined towards research in complex networks As Albert-Laacuteszloacute Barabaacutesi argues in his Commentary the past decade has indeed witnessed a lsquonetwork takeoverrsquo On the other hand James Crutchfield in his review of the tools for discovering patterns and quantifying their structural complexity demonstrates beautifully how fundamental theories of information and computation have led to a deeper understanding of just what lsquocomplex systemsrsquo are

For a topic as broad as complexity it is impossible to do justice to all of the recent developments The field has been shaped over decades by advances in physics engineering computer science biology and sociology and its ramifications are equally diverse But a selection had to be made and we hope that this Insight will prove inspiring and a showcase for the pivotal role that physicists are playing mdash and are bound to play mdash in the inherently multidisciplinary endeavour of making sense of complexity

Andreas Trabesinger Senior Editor

Complexity

COMMENTARYThe network takeoverAlbert-Laacuteszloacute Barabaacutesi 14

REVIEW ARTICLESBetween order and chaosJames P Crutchfield 17Communities modules and large-scale structure in networks M E J Newman 25Modelling dynamical processes in complex socio-technical systemsAlessandro Vespignani 32

PROGRESS ARTICLENetworks formed from interdependent networksJianxi Gao Sergey V Buldyrev H Eugene Stanley and Shlomo Havlin 40

C =maxP(X)

I [X Y ]

H (`)`

` i=1M

Msumi=1

KC(x i`)

KC(`)`

Pr(σ )sumx

Pr(σ )log2Pr(σ )

10|H 05|H05|T

05|T05|H10|T

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

C =maxP(X)

I [X Y ]

H (`)`

` i=1M

Msumi=1

KC(x i`)

KC(`)`

Pr(σ )sumx

Pr(σ )log2Pr(σ )

10|H 05|H05|T

05|T05|H10|T

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

C =maxP(X)

I [X Y ]

H (`)`

` i=1M

Msumi=1

KC(x i`)

KC(`)`

Pr(σ )sumx

Pr(σ )log2Pr(σ )

10|H 05|H05|T

05|T05|H10|T

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

C =maxP(X)

I [X Y ]

H (`)`

` i=1M

Msumi=1

KC(x i`)

KC(`)`

Pr(σ )sumx

Pr(σ )log2Pr(σ )

10|H 05|H05|T

05|T05|H10|T

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

C =maxP(X)

I [X Y ]

H (`)`

` i=1M

Msumi=1

KC(x i`)

KC(`)`

Pr(σ )sumx

Pr(σ )log2Pr(σ )

10|H 05|H05|T

05|T05|H10|T

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

C =maxP(X)

I [X Y ]

H (`)`

` i=1M

Msumi=1

KC(x i`)

KC(`)`

Pr(σ )sumx

Pr(σ )log2Pr(σ )

10|H 05|H05|T

05|T05|H10|T

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

` i=1M

Msumi=1

KC(x i`)

KC(`)`

Pr(σ )sumx

Pr(σ )log2Pr(σ )

10|H 05|H05|T

05|T05|H10|T

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

Pr(σ )sumx

Pr(σ )log2Pr(σ )

10|H 05|H05|T

05|T05|H10|T

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

H(16)16

Cmicro

hmicro

0 02 04 06 08 10

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

n = 4n = 3n = 2n = 1

n = 6n = 5

Cmicro

hmicro hmicro

0 02 04 06 08 100

0 02 04 06 08 10

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

σij =nijradickikj

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

[Aijminus

kikj2m

]δsisj

L=prodiltj

1minusAij

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

iinfin

1 β μ

sumiki that

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

sk =SkVk

ρs=sumk

P(k)sk

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

Nk =1Vk

sumi|ki=k

partNk

sumk prime

sumk primeP(k

pk =k〈k〉

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

Macroscopic level

Microscopic level

Mobility flows

Infectious

Susceptible

Subpop i

dcd = 0

Subpop j

Subpop i

Subpop j

d infin

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

pc pc p

Pinfin

First order

Second order

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

Pi(k)xk (1)

kPi(k)=partGi

Hi(x)equivsuminfin

k=0Pi(k)kxkminus1

〈k〉i=

Pinfini= pgi(p) (6)

pc=1〈k〉i

1gA(ψprime

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

Network A

Network B

B2A3 A5B6

B2 B3 B6

A3 A6A5

ψ prime1equiv p

τ+1 andφprime

τ =φprime

Attack

I stage

II stage

III stage

IV stage

Network A

Network B

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

10 20 30 40 50t

Theory

Simulation

qikqki

qi3qi2

xi= piKprodj=1

yij =xi

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

Pinfin= p

1minusp 1

n Pnminus1ninfin

(1minus(Pinfinp

) kminus1k

minus1

Pinfin=p2m

k(1minusq)m(21)

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

0 02 04 06 08 10p

0 05 1p p

P infinn = 1n = 2n = 5n = 10

q = 08

q = 02

02 04 06 08

q = 05

P infin

10a b c

xi= piKprodj=1

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

nphys2188pdf

References
nphys2190pdf
Complexities
Applications
Closing remarks
References
Acknowledgements
nphys2162pdf
Block models
Conclusion
References
Acknowledgements
nphys2160pdf
Conclusions
References
Acknowledgements
nphys2180pdf
References
Acknowledgements

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