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6 IEEE CIRCUITS AND SYSTEMS MAGAZINE 1531-636X/03/$17.002003
IEEE FIRST QUARTER 2003
In the past few years, the discovery ofsmall-world and
scale-free propertiesof many natural and artificial complexnetworks
has stimulated a great dealof interest in studying the
underlyingorganizing principles of various com-plex networks, which
has led to dra-matic advances in this emerging andactive field of
research. The presentarticle reviews some basic concepts,important
progress, and significantresults in the current studies of vari-ous
complex networks, with emphasison the relationship between
thetopology and the dynamics of suchcomplex networks. Some
fundamentalproperties and typical complex net-work models are
described; and, as anexample, epidemic dynamics are ana-lyzed and
discussed in some detail.Finally, the important issue of
robust-ness versus fragility of dynamicalsynchronization in complex
networksis introduced and discussed.
Index termscomplex network, small-world network, scale-free
network,synchronization, robustness
Xiao Fan Wang and Guanrong Chen
Complex Networks:
Small-World, Scale-Free and Beyond
Abstract
D
IGIT
AL
VIS
ION
, LT
D.
Feature
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Introduction
Complex networks are currently being studied acrossmany fields
of science [1-3]. Undoubtedly, manysystems in nature can be
described by models of
complex networks, which are structures consisting ofnodes or
vertices connected by links or edges. Examplesare numerous. The
Internet is a network of routers ordomains. The World Wide Web
(WWW) is a network ofwebsites (Fig. 1). The brain is a network of
neu-rons. An organization is a network of people.The global economy
is a network of nationaleconomies, which are themselves networks
ofmarkets; and markets are themselves networksof interacting
producers and consumers. Foodwebs and metabolic pathways can all be
repre-sented by networks, as can the relationshipsamong words in a
language, topics in a conver-sation, and even strategies for
solving a mathe-matical problem. Moreover, diseases aretransmitted
through social networks; and com-puter viruses occasionally spread
through theInternet. Energy is distributed through trans-portation
networks, both in living organisms,man-made infrastructures, and in
many physicalsystems such as the power grids. Figures 2-4
areartistic drawings that help visualize the com-plexities of some
typical real-world networks.
The ubiquity of complex networks in sci-ence and technology has
naturally led to a setof common and important research
problemsconcerning how the network structure facili-tates and
constrains the network dynamicalbehaviors, which have largely been
neglectedin the studies of traditional disciplines. Forexample, how
do social networks mediate thetransmission of a disease? How do
cascadingfailures propagate throughout a large powertransmission
grid or a global financial network?What is the most efficient and
robust architecture for aparticular organization or an artifact
under a changingand uncertain environment? Problems of this kind
areconfronting us everyday, problems which demandanswers and
solutions.
For over a century, modeling of physical as well asnon-physical
systems and processes has been performedunder an implicit
assumption that the interaction pat-terns among the individuals of
the underlying system orprocess can be embedded onto a regular and
perhapsuniversal structure such as a Euclidean lattice. In
late1950s, two mathematicians, Erds and Rnyi (ER), made
a breakthrough in the classical mathematical graph theo-ry. They
described a network with complex topology by arandom graph [4].
Their work had laid a foundation of therandom network theory,
followed by intensive studies inthe next 40 years and even today.
Although intuitionclearly indicates that many real-life complex
networks areneither completely regular nor completely random, theER
random graph model was the only sensible and rigor-
ous approach that dominated scientists thinking aboutcomplex
networks for nearly half of a century, due essen-tially to the
absence of super-computational power anddetailed topological
information about very large-scalereal-world networks.
In the past few years, the computerization of dataacquisition
and the availability of high computing powerhave led to the
emergence of huge databases on variousreal networks of complex
topology. The public access tothe huge amount of real data has in
turn stimulated greatinterest in trying to uncover the generic
properties of dif-ferent kinds of complex networks. In this
endeavor, two
7
Xiao Fan Wang is with the Department of Automation, Shanghai
Jiao Tong University, Shanghai 200030, P. R. China. Email:
[email protected]. Guanrong (Ron) Chen is with the Department of
Electronic Engineering and director of the Centre for Chaos Control
and Synchronization, CityUniversity of Hong Kong, 83 Tat Chee
Avenue, Kowloon, Hong Kong SAR, P. R. China. Email:
[email protected].
FIRST QUARTER 2003 IEEE CIRCUITS AND SYSTEMS MAGAZINE
Internet
WWW
Home Page
Domain 3
Domain 2
Router
Domain 1
Figure 1. Network structures of the Internet and the WWW. On the
Inter-net, nodes are routers (or domains) connected by physical
links such asoptical fibers. The nodes of the WWW are webpages
connected by direct-ed hyperlinks.
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significant recent discoveries are the small-world effectand the
scale-free feature of most complex networks.
In 1998, in order to describe the transition from a regu-lar
lattice to a random graph, Watts and Strogatz (WS)introduced the
concept of small-world network [5]. It isnotable that the
small-world phenomenon is indeed verycommon. An interesting
experience is that, oftentimes,soon after meeting a stranger, one
is surprised to find thatthey have a common friend in between; so
they both cheer:What a small world! An even more interesting
popularmanifestation of the small-world effect is the so-calledsix
degrees of separation principle, suggested by a socialpsychologist,
Milgram, in the late 1960s [6]. Although thispoint remains
controversial, the small-world pattern hasbeen shown to be
ubiquitous in many real networks. Aprominent common feature of the
ER random graph andthe WS small-world model is that the
connectivity distri-bution of a network peaks at an average value
and decaysexponentially. Such networks are called exponential
net-works or homogeneous networks, because each nodehas about the
same number of link connections.
Another significant recent discovery in the field of com-plex
networks is the observation that many large-scalecomplex networks
are scale-free, that is, their connectivitydistributions are in a
power-law form that is independentof the network scale [7, 8].
Differing from an exponentialnetwork, a scale-free network is
inhomogeneous in nature:most nodes have very few link connections
and yet a fewnodes have many connections.
The discovery of the small-world effect and scale-free
feature of complex networks has led to dramaticadvances in the
field of complex networks theory in thepast few years. The main
purpose of this article is to pro-vide some introduction and
insights into this emergingnew discipline of complex networks, with
emphasis onthe relationship between the topology and
dynamicalbehaviors of such complex networks.
Some Basic Concepts
Although many quantities and measures of complex net-works have
been proposed and investigated in the lastdecades, three
spectacular conceptsthe average pathlength, clustering coefficient,
and degree distributionplay a key role in the recent study and
development ofcomplex networks theory. In fact, the original
attempt ofWatts and Strogatz in their work on small-world
networks[5] was to construct a network model with small averagepath
length as a random graph and relatively large clus-tering
coefficient as a regular lattice, which evolved tobecome a new
network model as it stands today. On theother hand, the discovery
of scale-free networks wasbased on the observation that the degree
distributions ofmany real networks have a power-law form, albeit
power-law distributions have been investigated for a long time
inphysics for many other systems and processes. This sec-tion
provides a brief review of these important concepts.
Average Path LengthIn a network, the distance dij between two
nodes, labeledi and j respectively, is defined as the number of
edges
8 IEEE CIRCUITS AND SYSTEMS MAGAZINE FIRST QUARTER 2003
Large-Scale Organization
FunctionalModules
Regulatory Motifs
Genes
Information Storage Processing Execution
mRNA
Proteins Metabolites
ATP
Metabolic Pathways
Org
anis
m S
peci
ficity
Universality
Leu3
LEU1 BAT1 ILV2
ATP ATP ATPADP ADP ADP
UMP UDP UTP CTP
Mg2+ Mg2+Mg2+
Figure 2. [Courtesy of SCIENCE] A simple complexity pyra-mid
composed of various molecular components of cell-
genes, RNAs, proteins, and metabolites [47]. The bottomof the
pyramid shows the traditional representation of
the cells functional organization (level 1). There is
aremarkable integration of various layers at both the
regulatory and the structural levels. Insights intothe logic of
cellular organization can be gained
when one views the cell as an individual com-plex network in
which the components are
connected by functional links. At the low-est level, these
components form genet-
ic-regulatory motifs or metabolic path-ways (level 2), which in
turn are the
building blocks of functional mod-ules (level 3). Finally, these
mod-
ules are nested, generating ascale-free hierarchical archi-
tecture (level 4).
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along the shortest path connecting them. The diameter Dof a
network, therefore, is defined to be the maximal dis-tance among
all distances between any pair of nodes inthe network. The average
path length L of the network,then, is defined as the mean dis-tance
between two nodes, aver-aged over all pairs of nodes.Here, L
determines the effectivesize of a network, the mosttypical
separation of one pair ofnodes therein. In a friendshipnetwork, for
instance, L is theaverage number of friends exist-ing in the
shortest chain con-necting two persons in thenetwork. It was an
interestingdiscovery that the average pathlength of most real
complex net-works is relatively small, even inthose cases where
these kindsof networks have many feweredges than a typical
globallycoupled network with a equalnumber of nodes. This
small-ness inferred the small-worldeffect, hence the name of
small-world networks.
Clustering CoefficientIn your friendship network, it isquite
possible that your friendsfriend is also your direct friend;or, to
put it another way, two ofyour friends are quite possiblyfriends of
each other. This prop-erty refers to the clustering of thenetwork.
More precisely, one candefine a clustering coefficient C asthe
average fraction of pairs ofneighbors of a node that are
alsoneighbors of each other. Supposethat a node i in the network
has kiedges and they connect this nodeto ki other nodes. These
nodesare all neighbors of node i. Clear-ly, at most ki (ki 1)/2
edges canexist among them, and thisoccurs when every neighbor
ofnode i connected to every otherneighbor of node i. The
clusteringcoefficient C i of node i is thendefined as the ratio
between thenumber E i of edges that actually
exist among these ki nodes and the total possible numberki(ki
1)/2, namely, Ci = 2Ei/(ki(ki 1)). The clusteringcoefficient C of
the whole network is the average of Ci overall i. Clearly, C 1; and
C = 1 if and only if the network is
9FIRST QUARTER 2003 IEEE CIRCUITS AND SYSTEMS MAGAZINE
(a) Food Web (c) Social Network
(b) Metabolic Network (d) Java Network
Figure 3. Wiring diagrams for several complex networks. (a) Food
web of the Little RockLake shows who eats whom in the lake. The
nodes are functionally distinct trophicspecies. (b) The metabolic
network of the yeast cell is built up of nodesthe substratesthat
are connected to one another through links, which are the actual
metabolic reactions.(c) A social network that visualizes the
relationship among different groups of people inCanberra,
Australia. (d) The software architecture for a large component of
the JavaDevelopment Kit 1.2. The nodes represent different classes
and a link is set if there issome relationship (use, inheritance,
or composition) between two corresponding classes.
(a) (b)
Figure 4. [Courtesy of Richard V. Sole] Wiring diagrams of a
digital circuit (a), and an oldtelevision circuit (b). The dots
correspond to components, and the lines, wiring. Concentricrings
indicate a hierarchy due to the nested modular structure of the
circuits.
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10 IEEE CIRCUITS AND SYSTEMS MAGAZINE FIRST QUARTER 2003
Network Size Clustering coefficient Average path length Degree
exponent
Internet, domain level [13] 32711 0.24 3.56 2.1
Internet, router level [13] 228298 0.03 9.51 2.1
WWW [14] 153127 0.11 3.1 in = 2.1 out = 2.45E-mail [15] 56969
0.03 4.95 1.81
Software [16] 1376 0.06 6.39 2.5
Electronic circuits [17] 329 0.34 3.17 2.5
Language [18] 460902 0.437 2.67 2.7
Movie actors [5, 7] 225226 0.79 3.65 2.3
Math. co-authorship [19] 70975 0.59 9.50 2.5
Food web [20, 21] 154 0.15 3.40 1.13
Metabolic system [22] 778 3.2 in = out = 2.2
Table 1. Small-world pattern and scale-free property of several
real networks. Each network has the number of nodes N , the
clustering coeffi-cient C , the average path length L and the
degree exponent of the power-law degree distribution. The WWW and
metabolic networkare described by directed graphs.
is globally coupled, which means that every node in thenetwork
connects to every other node. In a completelyrandom network
consisting of N nodes, C 1/N , which isvery small as compared to
most real networks. It has beenfound that most large-scale real
networks have a tendencytoward clustering, in the sense that their
clustering coeffi-cients are much greater than O(1/N), although
they arestill significantly less than one (namely, far away
frombeing globally connected). This, in turn, means that mostreal
complex networks are not completely random. There-fore they should
not be treated as completely random andfully coupled lattices
alike.
Degree DistributionThe simplest and perhaps also the most
important char-acteristic of a single node is its degree. The
degree ki of anode i is usually defined to be the total number of
its con-nections. Thus, the larger the degree, the more impor-tant
the node is in a network. The average of ki over all iis called the
average degree of the network, and is denot-ed by < k >. The
spread of node degrees over a networkis characterized by a
distribution function P(k), which isthe probability that a randomly
selected node has exact-ly k edges. A regular lattice has a simple
degree sequencebecause all the nodes have the same number of
edges;and so a plot of the degree distribution contains a
singlesharp spike (delta distribution). Any randomness in
thenetwork will broaden the shape of this peak. In the limit-ing
case of a completely random network, the degreesequence obeys the
familiar Poisson distribution; and theshape of the Poisson
distribution falls off exponentially,
away from the peak value < k >. Because of this
expo-nential decline, the probability of finding a node with kedges
becomes negligibly small for k >> < k >.In the pastfew
years, many empirical results showed that for mostlarge-scale real
networks the degree distribution deviatessignificantly from the
Poisson distribution. In particular, fora number of networks, the
degree distribution can be bet-ter described by a power law of the
form P(k) k . Thispower-law distribution falls off more gradually
than anexponential one, allowing for a few nodes of very
largedegree to exist. Because these power-laws are free of
anycharacteristic scale, such a network with a power-lawdegree
distribution is called a scale-free network. Somestriking
differences between an exponential network and ascale-free network
can be seen by comparing a U.S.roadmap with an airline routing map,
shown in Fig. 5.
The small-world and scale-free features are common tomany
real-world complex networks. Table 1 shows someexamples that might
interest the circuits and systemscommunity (for example, the
discovery of the scale-freefeature of the Internet has motivated
the development ofa new brand of Internet topology generators
[9-12]).
Complex Network Models
Measuring some basic properties of a complex network,such as the
average path length L, the clustering coeffi-cient C , and the
degree distribution P(k), is the first steptoward understanding its
structure. The next step, then,is to develop a mathematical model
with a topology ofsimilar statistical properties, thereby obtaining
a plat-form on which mathematical analysis is possible.
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11FIRST QUARTER 2003 IEEE CIRCUITS AND SYSTEMS MAGAZINE
Regular Coupled NetworksIntuitively, a globally coupled network
has the smallestaverage path length and the largest clustering
coefficient.Although the globally coupled network model captures
thesmall-world and large-clustering properties of many
realnetworks, it is easy to notice its limitations: a globally
cou-pled network with N nodes has N(N 1)/2 edges, whilemost
large-scale real networks appear to be sparse, that is,most real
networks are not fully connected and their num-ber of edges is
generally of order N rather than N2.
A widely studied, sparse, and regular network model isthe
nearest-neighbor coupled network (a lattice), whichis a regular
graph in which every node is joined only by a
few of its neighbors. The term lattice here may suggesta
two-dimensional square grid, but actually it can havevarious
geometries. A minimal lattice is a simple one-dimensional
structure, like a row of people holdinghands. A nearest-neighbor
lattice with a periodic bound-ary condition consists of N nodes
arranged in a ring,where each node i is adjacent to its neighboring
nodes,i = 1, 2, , K/2, with K being an even integer. For a largeK ,
such a network is highly clustered; in fact, the cluster-ing
coefficient of the nearest-neighbor coupled network isapproximately
C = 3/4.
However, the nearest-neighbor coupled network is nota
small-world network. On the contrary, its average path
Poisson Distribution Power-Law Distribution
Exponential Network Scale-Free Network
k 1 10 100 1000
0.0001
0.001
0.01
1
0.1
P(k
)
P(k
)
k
Figure 5. [Courtesy of A.-L. Barabsi] Differences between an
exponential networka U.S. roadmap and a scale-free networkan
air-line routing map. On the roadmap, the nodes are cities that are
connected by highways. This is a fairly uniform network: each
majorcity has at least one link to the highway system, and there
are no cities served by hundreds of highways. The airline routing
map dif-fers drastically from the roadmap. The nodes of this
network are airports connected by direct flights among them. There
are a fewhubs on the airline routing map, including Chicago,
Dallas, Denver, Atlanta, and New York, from which flights depart to
almost allother U.S. airports. The vast majority of airports are
tiny, appearing as nodes with one or a few links connecting them to
one or sev-eral hubs.
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length is quite large and tends to infinity as N . Thismay help
explain why it is difficult to achieve any dynam-ical process
(e.g., synchronization) that requires globalcoordination in such a
locally coupled network. Doesthere exist a regular network that is
sparse and clustered,but has a small average path length? The
answer is Yes. Asimple example is a star-shaped coupled network,
inwhich there is a center node and each of the other N 1nodes only
connect to this center but not among them-selves. For this kind of
network, the average path lengthtends to 2 and its clustering
coefficient tends to 1, asN . The star-shaped network model
captures thesparse, clustering, small-world, as well as some
otherinteresting properties of many real-world networks.
There-fore, in this sense, it is better than the regular lattice as
amodel of many well-known real networks. Clearly, though,most real
networks do not have a precise star shape.
Random GraphsAt the opposite end of the spectrum from a
completelyregular network is a network with a completely
randomgraph, which was studied first by Erds and Rnyi (ER)about 40
years ago [4].
Try to imagine that you have a large number (N >> 1)of
buttons scattered on the floor. With the same probabil-ity p, you
tie every pair of buttons with a thread. The result
is a physical example of an ER random graph with N nodesand
about pN(N 1)/2 edges (Fig. 6). The main goal ofthe random graph
theory is to determine at what connec-tion probability p a
particular property of a graph willmost likely arise. A remarkable
discovery of this type wasthat important properties of random
graphs can appearquite suddenly. For example, if you lift up a
button, howmany other buttons will you pick up thereby? ER
showedthat, if the probability p is greater than a certain
thresholdpc (ln N)/N , then almost every random graph is
con-nected, which means that you will pick up all the buttonson the
floor by randomly lifting up just one button.
The average degree of the random graph is< k >= p(N 1) =
pN . Let Lrand be the average pathlength of a random network.
Intuitively, about < k >Lrand
nodes of the random network are at a distance Lrand orvery close
to it. Hence, N < k >Lrand , which means thatLrand ln N/ <
k >. This logarithmic increase in averagepath length with the
size of the network is a typical small-world effect. Because ln N
increases slowly with N , itallows the average path length to be
quite small even in afairly large network. On the other hand, in a
completelyrandom network, for example in your friendship
network(say it is completely random), the probability that two
ofyour friends are friends themselves is no greater than
theprobability that two randomly chosen persons from yournetwork
happen to be friends. Hence, the clustering coef-ficient of the ER
model is C = p =< k > /N
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WS Small-World Model Algorithm1) Start with order: Begin with a
nearest-neigh-
bor coupled network consisting of N nodesarranged in a ring,
where each node i is adjacent toits neighbor nodes, i = 1, 2, ,
K/2, with K beingeven.
2) Randomization: Randomly rewire each edgeof the network with
probability p; varying p in sucha way that the transition between
order (p = 0) andrandomness (p = 1) can be closely monitored.
Rewiring within this context means shifting one end ofthe
connection to a new node chosen at random from thewhole network,
with the constraints that any two differentnodes cannot have more
than one connection betweenthem, and no node can have a connection
with itself. Thisprocess introduces pN K/2 long-range edges, which
con-nect nodes that otherwise would be part of different
neigh-borhoods. Both the behaviors of the clustering coefficientC
(p) and of the average path length L(p) in the WS small-world model
can be considered as a function of therewiring probability p. A
regular ring lattice (p = 0) is high-ly clustered (C (0) = 3/4) but
has a large average pathlength (L(0) = N2K >> 1). It is found
that, for a small prob-ability of rewiring, when the local
properties of the networkare still nearly the same as those for the
original regularnetwork, and when the clustering coefficient does
not dif-fer subsequently from its initial value (C (p) C (0)),
theaverage path length drops rapidly and is in the same orderas the
one for random networks (L(p) >> L(0)) (Fig. 8).This result
is actually quite natural. On the one hand, it issufficient to make
several random rewirings to decrease
the average path length significantly. On theother hand, several
rewired links cannotcrucially change the local clustering proper-ty
of the network.
The small-world model can also beviewed as a homogeneous
network, inwhich all nodes have approximately thesame number of
edges. In this regard, theWS small-world network model is similar
tothe ER random graph model. The work onWS small-world networks has
started anavalanche of research on new models ofcomplex networks,
including some variantsof the WS model. A typical variant was
theone proposed by Newman and Watts [23],referred to as the NW
small-world modellately. In the NW model, one does not breakany
connection between any two nearestneighbors, but instead, adds with
probabil-ity p a connection between a pair of nodes.Likewise, here
one does not allow a node to
be coupled to another node more than once, or to couplewith
itself. With p = 0, the NW model reduces to the origi-nal
nearest-neighbor coupled network, and if p = 1 itbecomes a globally
coupled network. The NW model issomewhat easier to analyze than the
original WS modelbecause it does not lead to the formation of
isolated clus-ters, whereas this can indeed happen in the WS model.
Forsufficiently small p and sufficiently large N , the NW modelis
essentially equivalent to the WS model. Today, thesetwo models are
together commonly termed small-worldmodels for brevity.
The small-world models have their roots in social net-works,
where most people are friends with their immedi-ate neighbors, for
example neighbors on the same street
13FIRST QUARTER 2003 IEEE CIRCUITS AND SYSTEMS MAGAZINE
Rewiring of Links
P=0 0
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or colleagues in the same office. On the other hand, manypeople
have a few friends who are far away in distance,such as friends in
other countries, which are representedby the long-range edges
created by the rewiring proce-dure in the WS model, or by the
connection-adding pro-cedure in the NW model.
Scale-Free ModelsA common feature of the ER random graph and the
WSsmall-world models is that the connectivity distribution ofthe
network is homogenous, with peak at an average valueand decay
exponentially. Such networks are called expo-nential networks. A
significant recent discovery in the fieldof complex networks is the
observation that a number oflarge-scale complex networks, including
the Internet,WWW, and metabolic networks, are scale-free and
theirconnectivity distributions have a power-law form.
To explain the origin of power-law degree distribution,Barabsi
and Albert (BA) proposed another networkmodel [7,8]. They argued
that many existing models failto take into account two important
attributes of most realnetworks. First, real networks are open and
they aredynamically formed by continuous addition of new nodesto
the network; but the other models are static in thesense that
although edges can be added or rearranged,the number of nodes is
fixed throughout the formingprocess. For example, the WWW is
continually sproutingnew webpages, and the research literature
constantlygrows since new papers are continuously being pub-lished.
Second, both the random graph and small-worldmodels assume uniform
probabilities when creating newedges, but this is not realistic
either. Intuitively, webpages
that already have many links (such as the homepage ofYahoo or
CNN) are more likely to acquire even more links;a new manuscript is
more likely to cite a well-known andthus much-often-cited paper
than many other less-knownones. This is the so-called rich get
richer phenomenon,for which the other models do not account.
The BA model suggests that two main ingredients
ofself-organization of a network in a scale-free structure
aregrowth and preferential attachment. These point to thefacts that
most networks continuously grow by the addi-tion of new nodes, and
new nodes are preferentiallyattached to existing nodes with large
numbers of connec-tions (again, rich get richer). The generation
scheme ofa BA scale-free model is as follows:
BA Scale-Free Model Algorithm1) Growth: Start with a small
number (m0) of
nodes; at every time step, a new node is introducedand is
connected to m m0 already-existing nodes.
2) Preferential Attachment: The probability ithat a new node
will be connected to node i (one ofthe m already-existing nodes)
depends on thedegree ki of node i, in such a way that i = ki/
j kj.
After t time steps, this algorithm results in a networkwith N =
t + m0 nodes and mt edges (Fig. 9). Growingaccording to these two
rules, the network evolves into ascale-invariant state: The shape
of the degree distributiondoes not change over time, namely, does
not change dueto further increase of the network scale. The
correspon-ding degree distribution is described by a power law
withexponent 3, that is, the probability of finding a nodewith k
edges is proportional to k3.
Numerical results have indicated that, in comparisonwith a
random graph having the same size and the sameaverage degree, the
average path length of the scale-freemodel is somewhat smaller, and
yet the clustering coeffi-cient is much higher. This implies that
the existence of afew big nodes with very large degrees (i.e., with
a verylarge number of connections) plays a key role in bringingthe
other nodes of the network close to each other. How-ever, there is
today no analytical prediction formula forthe average path length
and the clustering coefficient forthe scale-free model. The BA
model is a minimal modelthat captures the mechanisms responsible
for the power-law degree distribution. This model has some evident
lim-itations when compared with some real-world networks.This
observation has in effect spurred more research onevolving
networks, with the intention to overcome limita-tions such as those
of the BA model. A summary of thesemodels is given by Albert and
Barabsi [2].
Recently, Milo et al. [24] defined the so-called net-work motifs
as patterns of interconnections occurring in
14 IEEE CIRCUITS AND SYSTEMS MAGAZINE FIRST QUARTER 2003
Figure 9. A scale-free network of 130 nodes, generated bythe BA
scale-free model. The five biggest nodes are shown inred, and they
are in contact with 60% of other nodes (green).
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complex networks at numbers that are significantly high-er than
those in completely random networks. Suchmotifs have been found in
networks ranging from bio-chemistry, neurobiology, ecology, to
engineering. Thisresearch may uncover the basic building blocks
pertain-ing to each class of networks.
Achilles Heel of Complex Networks
An interesting phenomenon of complex networks is theirAchilles
heelrobustness versus fragility. For illustra-tion, let us start
from a large and connected network. Ateach time step, remove a node
(Fig. 10). The disappear-ance of the node implies the removal of
all of its connec-tions, disrupting some of the paths among the
remainingnodes. If there were multiple paths between two nodes iand
j, the disruption of one of them may mean that thedistance dij
between them will increase, which, in turn,may cause the increase
of the average path length L ofthe entire network. In a more severe
case, when initiallythere was a single path between i and j, the
disruption ofthis particular path means that the two nodes
becomedisconnected. The connectivity of a network is robust
(orerror tolerant) if it contains a giant cluster comprisingmany
nodes, even after a removal of a fraction of nodes.
The predecessor of the Internetthe ARPANETwas
created by the US Department of Defense, by itsAdvanced Research
Projects Agency (ARPA), in the late1960s. The goal of the ARPANET
was to enable continuoussupply of communications services, even in
the case thatsome subnetworks and gateways were failing. Today,
theInternet has grown to be a huge network and has playeda crucial
role in virtually all aspects of the world. One maywonder if we can
continue to maintain the functionality ofthe network under
inevitable failures or frequent attacksfrom computer hackers. The
good news is that by ran-domly removing certain portions of domains
from theInternet, we have found that, even if more than 80% of
thenodes fail, it might not cause the Internet to collapse.However,
the bad news is that if a hacker targeted somekey nodes with very
high connections, then he couldachieve the same effect by
purposefully removing a verysmall fraction of the nodes (Fig. 11).
It has been shownthat such error tolerance and attack vulnerability
aregeneric properties of scale-free networks (Fig. 12)
[25-28].These properties are rooted in the extremely inhomoge-neous
nature of degree distributions in scale-free net-works. This attack
vulnerability property is called anAchilles heel of complex
networks, because the mytho-logical warrior Achilles had been
magically protected inall but one small part of his bodyhis
heel.
15FIRST QUARTER 2003 IEEE CIRCUITS AND SYSTEMS MAGAZINE
A A
AA
B B
BB
C C
CC
D D
DD
Remove r1
Remove r2Remove r2
Remove r1
r1
r1
r2r2
Figure 10. Illustration of the effects of node removal on
aninitially connected network. In the unperturbed state,dAB = dC D
= 2. After the removal of node r1 from the originalnetwork, dAB =
8. After the removal of node r2 from the orig-inal network, dC D =
7. After the removal of nodes r1 and r2,the network breaks into
three isolated clusters anddAB = dC D = .
Internet1.0
0.8
0.6
0.4
0.2
0.0
S
0.0 0.2 0.4 0.6 0.8 1.0
Figure 11. The relative size S of the largest cluster in
theInternet, when a fraction f of the domains are removed [25]. ,
random node removal; , preferential removal of themost connected
nodes.
Robustto RandomFailures Fragile
to IntentionalAttacks
Figure 12. The robust, yet fragile feature of complex
net-works.
-
Epidemic Dynamics in Complex Networks
For one specific example, the AIDS propagation networkis quite
typical. When AIDS first emerged as a diseaseabout twenty years
ago, few people could have predictedhow the epidemic would evolve,
and even fewer couldhave been able to describe with certainty the
best way offighting it. Unfortunately, according to estimates from
theJoint United Nations Programme on HIV/AIDS (UNAIDS)and the World
Health Organization (WHO), 21.8 millionpeople around the world had
died of AIDS up to the endof 2000 and 36.1 million people were
living with thehuman immunodeficiency virus (HIV) by the same
time.
As another example, in spite of technological progressand great
investments to ensure a secure supply of elec-tric energy,
blackouts of the electric transmission grid arenot uncommon.
Cascading failures in large-scale electricpower transmission
systems arean important cause of the cata-strophic blackouts. Most
wellknown is the cascading series offailures in power lines in
August1996, leading to blackouts in 11US states and two
Canadianprovinces. This incident leftabout 7 million customers
with-out power for up to 16 hours, andcost billions of dollars in
totaldamage. There is an urgent needfor developing
innovativemethodologies and conceptualbreakthroughs for analysis,
plan-ning, operation, and protection ofthe complex and dynamical
elec-tric power networks. In yet anoth-er example, the ILOVEYOU
computer virus spread overthe Internet in May 2000 and caused a
loss of nearly 7 bil-lion dollars in facility damage and computer
down-time.
How do diseases, jokes, and fashions spread out overthe social
networks? How do cascading failures propagatethrough large-scale
power grids? How do computer virus-es spread out through the
Internet? Serious issues likethese are attracting much attention
these days. Clearly,the topology of a network has great influence
on the over-all behavior of an epidemic spreading in the
network.Recently, some researchers have started to study
suchspreading phenomena, for example on small-world andscale-free
networks [29-34].
A notable attempt of Pastor-Satorras and Vespignani[31-32] was
to study both analytically and numerically alarge-scale dynamical
model on the spreading of epi-demics in complex networks. The
standard susceptible-infected-susceptible (SIS) epidemiological
model wasused for investigation. Each node of the network
repre-
sents an individual, and each link is a connection alongwhich
the infection can spread from one individual tosome others. It is
natural to assume that each individualcan only exist in one of two
discrete statessusceptibleand infected. At every time step, each
susceptible node isinfected with probability if it is connected to
at leastone infected node. At the same time, infected nodes
arecured and become again susceptible with probability .They
together define an effective spreading rate, = /.The updating can
be performed with both parallel andsequential dynamics. The main
prediction of the SISmodel in homogeneous networks (including
lattices, ran-dom graphs, and small-world models) is the presence
ofa nonzero epidemic threshold, c > 0. If c, the infec-tion
spreads and becomes persistent in time; yet if < c,the infection
dies out exponentially fast (Fig. 13 (a)).
It was found [31-32] that, while for exponential net-works the
epidemic threshold is a positive constant, for alarge class of
scale-free networks the critical spreadingrate tends to zero (Fig.
13(b)). In other words, scale-freenetworks are prone to the
spreading and the persistenceof infections, regardless of the
spreading rate of the epi-demic agents. It implies that computer
viruses can spreadfar and wide on the Internet by infecting only a
tiny frac-tion of the huge network. Fortunately, this is balanced
byexponentially small prevalence and by the fact that it istrue
only for a range of very small spreading rates(
-
waves, spiral waves, and spatiotemporal chaos. Also,
thesenetworks are important in modeling many large-scale real-world
systems.
In the past decade, special attention has been focusedon the
synchronization of chaotic dynamical systems. Forthe same reason,
many scientists have started to considerthe synchronization
phenomenon in large-scale networksof coupled chaotic oscillators.
These networks are usuallydescribed by systems of coupled ordinary
differentialequations or maps, with completely regular
topologicalstructures such as chains, grids, lattices, and globally
cou-pled graphs. Two typical settings are the discrete-time
cou-pled map lattice (CML) [35] and the continuous-time cellu-lar
neural (or more generally, nonlinear) networks (CNN)[36]. The main
advantage of these simple architectures isthat it allows one to
focus on the complexity caused by thenonlinear dynamics of the
nodes without worrying aboutadditional complexity in the network
structure; and anoth-er appealing feature is the ease of their
implementation byintegrated circuits.
The topology of a network, onthe other hand, often plays a
crucialrole in determining its dynamicalbehaviors. For example,
although astrong enough diffusive couplingwill result in
synchronization withinan array of identical nodes [37], itcannot
explain why many real-world complex networks exhibit astrong
tendency toward synchro-nization even with a relatively
weakcoupling. As an instance, it wasobserved that the apparently
inde-pendent routing messages from dif-ferent routers in the
Internet caneasily become synchronized, whilethe tendency for
routers towardssynchronization may depend heavi-ly on the topology
of the Internet[38]. One way to break up theunwanted
synchronization is foreach router to add a (sufficientlylarge)
component randomly to theperiod between two routing mes-sages.
However, the tendency tosynchronization in the Internet is sostrong
that changing one determin-istic protocol to correct the
syn-chronization is likely to generateanother synchrony elsewhere
at thesame time. This suggests that amore efficient solution
requires abetter understanding of the nature
of the synchronization behavior in such complex net-works as the
Internet.
Recently, synchronization in different small-world andscale-free
dynamical network models has been carefullystudied [39-45]. These
studies may shed new light on thesynchronization phenomenon in
various real-world com-plex networks.
A Typical Dynamical Network ModelConsider a typical dynamical
network consisting of Nidentical linearly and diffusively coupled
nodes, witheach node being an n-dimensional dynamical system(e.g.,
a chaotic system). The state equations of this net-work are
described by
xi = f(xi) + cN
j=1aijxj, i = 1, 2, , N. (1)
In this model, xi = (xi1, xi2, , xin)T n are the statevariables
of node i, the constant c > 0 represents the cou-
17FIRST QUARTER 2003 IEEE CIRCUITS AND SYSTEMS MAGAZINE
100080
60
40
20
0
500
400
300
200
100
0
80060040020010.5p
(a) (b)
N0
2sw 2sw
Figure 14. The second-largest eigenvalue 2sw of the coupling
matrix of the small-worldnetwork (1) [41]. (a) 2sw as a function of
the adding probability p with the network sizeN = 500. (b) 2sw as a
function of the network size with adding probability p = 0.1.
Synchronizability
Locally Coupled Small-World(a) (b)
-
pling strength, and nn is a constant 0 1 matrixlinking coupled
variables. If there is a connection betweennode i and node j (i =
j), then aij = aji = 1; otherwise,aij = aji = 0 (i = j). Moreover,
aii = ki, where ki is thedegree of node i. The coupling matrix A =
(aij) NNrepresents the coupling configuration of the network.
Dynamical network (1) is said to be (asymptotically)synchronized
if
x1(t) = x2(t) = = xN (t) = s(t), as t , (2)
where s(t) n is a solution of an isolated node, i.e.,s(t) =
f(s(t)). Here, s(t) can be an equilibrium point, aperiodic orbit,
or a chaotic attractor, depending on theinterest of the study.
Consider the case that the network is connected inthe sense that
there are no isolated clusters. Then, thecoupling matrix A =
(aij)NN is a symmetric irreduciblematrix. In this case, it can be
shown that 1 = 0 is thelargest eigenvalue of A with multiplicity 1
but all theother eigenvalues of A are strictly negative. Let 2 <
0be the second-largest eigenvalue of A. It has beenproved [40, 41]
that the synchronization state (2) isexponentially stable if
c d/2
, (3)
where d < 0 is a constant determined by the dynamics ofan
isolated node and the inner linking structural matrix .(In fact, d
can be more precisely characterized by the Lya-punov exponents of
the network [46].)
Given the dynamics of an isolated node and the innerlinking
structural matrix , the synchronizability of thedynamical network
(1) with respect to a specific couplingconfiguration A is said to
be strong if the network can syn-chronize with a small value of the
coupling strength c. Theabove result implies that the
synchronizability of thedynamical network (1) can be characterized
by the sec-ond-largest eigenvalue of its coupling matrix.
The second-largest eigenvalue of the coupling matrix of
aglobally coupled network is N, which implies that for anygiven and
fixed nonzero coupling strength c, a globally cou-pled network will
synchronize as long as its size N is largeenough. On the other
hand, the second-largest eigenvalue ofthe coupling matrix of a
nearest-neighbor coupled networktends to zero as N , which implies
that for any given andfixed nonzero coupling strength c, a
nearest-neighbor couplednetwork cannot synchronize if its size N is
sufficiently large.
Synchronization in Small-World NetworksConsider the dynamical
network (1) with NW small-worldconnections [41]. Let 2sw be the
second-largest eigenval-ue of the network coupling matrix. Figures
14 (a) and (b)
show the numerical values of 2sw as a function of theadding
probability p and the network size N , respectively.It can be seen
that, for any given coupling strength c > 0:(i) for any N >
|d|/c, there exists a critical value p suchthat if p p 1 then the
small-world network will syn-chronize; (ii) for any given p (0, 1],
there exists a criticalvalue N such that if N > N then the
small-world networkwill synchronize. These results imply that the
ability toachieve synchronization in a large-size
nearest-neighborcoupled network can be greatly enhanced by just
adding atiny fraction of distant links, thereby making the
networkbecome a small-world model. This reveals an advantage
ofsmall-world networks for achieving synchronization, ifdesired
(Fig. 15).
Synchronization in Scale-Free NetworksNow consider the dynamical
network (1) with BA scale-free connections instead [42]. Figure 16
shows that thesecond-largest eigenvalue of the corresponding
cou-pling matrix is very close to 1, which actually is
thesecond-largest eigenvalue of the star-shaped couplednetwork.
This implies that the synchronizability of a
18 IEEE CIRCUITS AND SYSTEMS MAGAZINE FIRST QUARTER 2003
2sf
0.90
0.85
0.95
1.00
200 400 600 800 1,000N
Figure 16. The second-largest eigenvalue of the couplingmatrix
of the scale-free network (1), for m0 = m = 3 ();m0 = m = 5( ); and
m0 = m = 7() [42].
Scale-Free Star-Shaped
Synchronizability
Hub
(a) (b)
Figure 17. Synchronizability of a scale-free network is aboutthe
same as that of a star-shaped coupled network.
-
scale-free network is about the same as that of a star-shaped
coupled network (Fig. 17). It may be due to theextremely
inhomogeneous connectivity distribution ofsuch networks: a few hubs
in a scale-free network playa similar (important) role as a single
center in a star-shaped coupled network.
The robustness of synchronization in a scale-freedynamical
network has also been investigated, againsteither random or
specific removal of a small fraction f(0 < f
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Xiao Fan Wang (M00) received the B.Sc.degree from Suzhou
University in 1986,the M.Sc. degree from Nanjing NormalUniversity
in 1991, and the Ph.D. degreefrom Southeast University in 1996.
FromOctober 1996 to August 1999 he was aPost-Doctorate Fellow and
then an Asso-
ciate Professor in the Department of Automation, Nan-jing
University of Scientific and Technology. FromSeptember 1999 to
August 2000, he was a Research Asso-ciate in the City University of
Hong Kong. From October2000 to January 2001, he was a
Post-Doctorate ResearchFellow in the University of Bristol, UK.
Currently he is aProfessor in the Department of Automation,
ShanghaiJiao Tong University. His current research interestsinclude
control and synchronization of complex dynami-cal systems and
networks.
Guanrong Chen (M87, SM92, F96)received the M.Sc. degree in
ComputerScience from Sun Yatsen (Zhongshan) Uni-versity and the
Ph.D. degree in AppliedMathematics from Texas A&M
University.Currently he is a Chair Professor and theDirector of the
Centre for Chaos Control
and Synchronization at the City University of Hong Kong.He is a
Fellow of the IEEE for his fundamental contributionsto the theory
and applications of chaos control and bifur-cation analysis.
Prof. Chen has numerous publications since 1981 in thefields of
nonlinear systems, in both dynamics and controls.Among his
publications are the research monographs andedited books entitled
Hopf Bifurcation Analysis: A Frequen-cy Domain Approach (World
Scientific, 1996), From Chaos toOrder: Methodologies, Perspectives
and Applications (WorldScientific, 1998), Controlling Chaos and
Bifurcations in Engi-neering Systems (CRC Press, 1999), and Chaos
in Circuitsand Systems (World Scientific, 2002).
Prof. Chen served and is serving as the Advisory Edi-tor,
Features Editor, and Associate Editor for 7 interna-tional journals
including the IEEE Transactions andMagazine on Circuits and Systems
and the InternationalJournal of Bifurcation and Chaos. He received
the 1998Harden-Simons Outstanding Prize for the Best JournalPaper
Award from the American Society of EngineeringEducation, the 2001
M. Barry Carlton Best Annual Trans-actions Paper Award from the
IEEE Aerospace and Elec-tronic Systems Society, and the 2002 Best
Paper Awardfrom the Institute of Information Theory and
Automation,Academy of Sciences of the Czech Republic. He is
Hon-orary Professor of the Central Queensland University,Australia,
as well as Honorary Guest-Chair Professors ofseveral universities
in China.
20 IEEE CIRCUITS AND SYSTEMS MAGAZINE FIRST QUARTER 2003
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