An Introduction to Complex Systems Science - unibo.it · Introductory concepts Boolean networks Complex networks An Introduction to Complex Systems Science Andrea Roli DEIS, Campus

Introductory conceptsBoolean networksComplex networks

An Introduction to Complex Systems Science

Andrea Roli

DEIS, Campus of CesenaAlma Mater Studiorum Universita di Bologna

[email protected]

Andrea Roli An Introduction to Complex Systems Science


DisclaimerThe field of Complex systems science is wide and it involvesnumerous themes and disciplines.This talk just provides an informal introduction to some relevanttopics in this area.



Outline

1 Introductory conceptsComplex systemsMain concepts

2 Boolean networksBasicsRandom Boolean NetworksApplications of Boolean Networks

3 Complex networks



Complex systemsMain concepts

Outline



3 Complex networks




Complex systems science

CSSA new field of science studying how parts of a system give riseto the collective behaviours of the system, and how the systeminteracts with its environment.

It focuses on certain questions about parts, wholes andrelationships.




Complex systems

Examples of complex systems are:

The brainThe societyThe ecosystemThe cellThe ant coloniesThe stock market. . .





CSS is interdisciplinary and it involves:

MathematicsPhysicsComputer scienceBiologyEconomyPhilosophy

...just to mention some.





Three main interrelated approaches to the modern study ofcomplex systems:

1 How interactions give rise to patterns of behaviour

2 Understanding the ways of describing complex systems

3 Understanding the process of formation of complexsystems through pattern formation and evolution





Some prominent research topics in CSS:

Evolution & emergenceSystems biologyInformation & computationComplex networksPhysics of Complexity




Reductionism vs. Holism

Reductionism: an approach to understanding the nature ofcomplex things by reducing them to the interactions of theirparts.

Holism: idea that all the properties of a system cannot bedetermined or explained by its component parts alone.Summarised with the sentence The whole is more than thesum of its parts.




Complex vs. Complicated

Complex: from Latin (cum + plexere); it means “intertwined”.

Complicated: from Latin (cum + plicare); it means “foldedtogether”.




Properties of complex systems

Complex systems enjoy (some of) these properties:

Composed of many elementsNonlinear interactionsNetwork topologyPositive and negative feedbacksAdaptive and evolvableRobustLevels of organisation




Outline



3 Complex networks




Emergence

Emergence refers to understanding how collective propertiesarise from the properties of parts.

A common case of emergence is self-organisation




Self-organisation

Dynamical mechanisms whereby structures appear at the globallevel from interactions among lower-level components.

Creation of spatio-temporal structuresPossible coexistence of several stable states (multistability)Existence of bifurcations when some parameters arevaried




Example: Benard cells




Further examples




Model of a system

ModelA model is an abstract and schematic representation of asystem. It is also usually a formal representation of the system.

It makes it possible to:

investigate some properties of the systemmake predictions on the future

It is usually in the form of a set of objects and the relationsamong them




Properties of a model

It represents only a portion of the systemIt only captures some of the system’s featuresThe abstraction process involves simplification,aggregation and omission of details




Example: the logistic map

xt+1 = rxt(1− xt)

xi ∈ [0,1]r ∈ [0,+∞[

Simple model of population growthDifferent kinds of behaviour depending on the values of r




Logistic map: steady states

r ≤ 3→ single value3 < r < 3.57→ repeated sequence of valuesr ≥ 3.57→ sequence of values without apparent structure




Attractors

AttractorPortion of the state space towards which a dynamical systemevolves over time.

Fixed point(Limit) CycleStrange attractor




Logistic map: attractors

r ≤ 3→ fixed point3 < r < 3.57→ cycler ≥ 3.57→ strange attractor




Deterministic chaos

Deterministic modelSensitivity to initial conditionsIn practice, it is impossible to make long term predictionsThe attractor is a strange attractor




Strange attractor

A strange attractor is a fractalNon-integer dimensionSelf-similarity




Complexity

Complexity lies at the edge of order and chaos




Complexity

Statistical complexity of a systemComplexity = Entropy × Disequilibrium



BasicsRandom Boolean NetworksApplications of Boolean Networks

Outline



3 Complex networks




Boolean networks

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Boolean networks

Introduced by Stuart Kauffman in 1969 as a geneticregulatory network model

Discrete-time / discrete-state dynamical system

Non trivial (complex) dynamics




Structure

Oriented graph of N nodes

Node i :

- Boolean value xi- Boolean function fi

Boolean function arguments arevariables associated to input nodesof iNode state (i.e., Boolean variable)updated as a function of fi




Structure

Oriented graph of N nodesNode i :






Structure


- Boolean value xi

- Boolean function fiBoolean function arguments arevariables associated to input nodesof iNode state (i.e., Boolean variable)updated as a function of fi

X 1

X 2

X 3




Structure




X 1

X 2

X 3

AND

OR

OR




Structure




X 1

X 2

X 3

AND

OR

OR




Dynamics

System state at time t : s(t) = (x1(t), . . . , xN(t))Dynamics controls node updateSynchronous vs. asynchronous dynamics

Synchronous dynamics (and deterministic update rules):

One successor per stateCardinality of state space 2N




Dynamics

System state at time t : s(t) = (x1(t), . . . , xN(t))Dynamics controls node updateSynchronous vs. asynchronous dynamics

Synchronous dynamics (and deterministic update rules):

One successor per stateCardinality of state space 2N




DynamicsTransition function

t t + 1x1 x2 x3 x1 x2 x3

0 0 0 0 0 00 0 1 0 1 00 1 0 0 0 10 1 1 1 1 11 0 0 0 1 11 0 1 0 1 11 1 0 0 1 11 1 1 1 1 1

X 1

X 2

X 3

AND

OR

OR




DynamicsTrajectory in state space

100

011110

101

111

001 010

000





Trajectory composed of two parts:TransientAttractor

Attractors:Fixed pointsCycles

100

011110

101

111

001 010

000





Basin of attraction of A:

set of states belonging to the trajectoryending at attractor A

100

011110

101

111

001 010

000




Dynamics




Why are BNs interesting?

Minimal complex system

Several important phenomena in genetics can bereproduced

Tight connections with the satisfiability problem




Outline



3 Complex networks




Random Boolean networksModel

K inputs per node

Inputs chosen at random, no self-arcs

Random Boolean functions: each entry of truth table hasprobability p = 0.5 of being set to 1




Random Boolean networksProperties

K = 1: ORDER

Frozen dynamicsExtremely robust: small perturbations die out quickly





K ≥ 3: (pseudo) CHAOS

Very long cycles (∼ 2N )Sensitivity to initial conditionsNot robust: small perturbations spread quickly throughoutthe system





K = 2: CRITICALITY

Short cycles (∼ low degree polynomial of N)Robust: small perturbations die out (in the long term) orkeep smallSecond order phase transition




Critical parameters

From the theory:

Kc = [2pc(1− pc)]−1




Extensions and variants

AsynchronousProbabilisticMultivalued logicsContinuous variables ruled by differential equations (e.g.,Glass networks)Multiple interacting BNs




Outline



3 Complex networks




BNs in biology

Cellular dynamics models

Models of specific genetic regulatory networks

Cancer and stem cell models




Cell dynamics

Main resultAttractor (or set-of)↔ cell type





Cell dynamics

Main resultAttractor (or set-of)↔ cell type





Robustness & adaptiveness

Main resultCritical BNs make it possible to achieve the best balancebetween robustness and adaptiveness.

Real cells are critical




Robustness & adaptiveness

Main resultCritical BNs make it possible to achieve the best balancebetween robustness and adaptiveness.

Real cells are critical




KO avalanches

Main resultBNs can reproduce the same avalanche distribution as realgenetic networks.

Same results with several kinds of BNs (e.g., Glass nets)




KO avalanches

Main resultBNs can reproduce the same avalanche distribution as realgenetic networks.

Same results with several kinds of BNs (e.g., Glass nets)




BNs in engineering and computer science

Satisfiability problem

Learning systems

Boolean network robotics




Boolean network roboticshttp://iridia.ulb.ac.be/bn-robotics/

Dynamical system theory and complexity science are richsources for:

analysing artificial agents and robotsdesign principles and guidelines

Boolean network roboticsBoolean network robotics concerns the use of Boolean net-works, and other models from complex systems science, asrobot programs.




Boolean network roboticshttp://iridia.ulb.ac.be/bn-robotics/

Dynamical system theory and complexity science are richsources for:

analysing artificial agents and robotsdesign principles and guidelines

Boolean network roboticsBoolean network robotics concerns the use of Boolean net-works, and other models from complex systems science, asrobot programs.



Complex networks



Complex networks

System’s behaviour depends on the structure of relationamong the components

Useful models from graph theory

Recent research stream in CSS



Graph as a structure model

Key ideas:

Represent the entities of the system as graph vertices(nodes)

Represent the relations between entities as edges (arcs)

A vertex can be a single element or a sub-system



Examples of networks

Technological nets: Internet, telephone, power grids,transportation, etc.

Social nets: friendship, collaboration, etc.

Nets of information: WWW, citations, tec.

Biological nets: biochemical, neural, ecological, etc.



Random graphs

First model of the topology of a complex systemInteresting theoretical resultsBaseline for comparison with other topologies

Strictly speaking, a random graph model is defined in terms ofan ensemble of graphs generated through a given procedure:

vertices are positioned by choosing two vertices at random(i.e., on the basis of a uniform distribution)

→ degree distribution is Poissonian (Gaussian in the limit case)



Random graphs







Random graphs







Random graphs



Features of interest

Vertex degree (in- and out-degree if edges are oriented)

Diameter, characteristic path length et similia

Clustering coefficient



Characteristic length L(G)

Informally: average path length between any pair of vertices.

Random graphs→ short L(G)

Grid graphs→ long L(G)



Characteristic length L(G)

Informally: average path length between any pair of vertices.

Random graphs→ short L(G)

Grid graphs→ long L(G)



Clustering γ

c

a

b

Informally: γ quantifies the probability that,given vertex a connected to b and c, there isan edge between b and c.

Random graphs→ low γ

Grid graphs→ high γ



Clustering γ

c

a

b

Informally: γ quantifies the probability that,given vertex a connected to b and c, there isan edge between b and c.

Random graphs→ low γ

Grid graphs→ high γ



Scale-free nets

Scale-free networks can represent the topology of:

Social relations (e.g.,scientific collaborations)Web-pagesThe Internet. . .



Scale-free nets

Degree distribution:number of vertices with degree k ∼ k−γ

Few vertices with many connections (hubs) and manyvertices with few connectionsRobust against accidental damagesFragile w.r.t. specific attacks



Scale-free nets



Scale-free nets

Dynamics dramatically different from random and regulartopologiesImplications in medicine (e.g., epidemics), society, InternetRelated to small-world phenomena (low length, highclustering)



Scale-free nets development

A net evolves to a scale-free topology if the following twoconditions hold (sufficient condition):

Growth: older vertices have a higher number ofconnectionsPreferential attachment : new vertices tend to be attachedto vertices with many connections (prob. is proportional tothe number of links)

Model variants taking also into account vertex fitness



References

Serra, R., Zanarini, G.: Complex Systems and CognitiveProcesses. Springer, Berlin, Germany (1990)

Bar–Yam, Y.: Dynamics of Complex Systems. Studies innonlinearity, Addison–Wesley, Reading, MA (1997)

Kauffman, S.: The Origins of Order: Self-Organization andSelection in Evolution. Oxford University Press, UK (1993)

Newman, M.E.J.: Networks. An Introduction. OxfordUniversity Press, UK (2010)


An Introduction to Complex Systems Science - unibo.it · Introductory concepts Boolean networks Complex networks An Introduction to Complex Systems Science Andrea Roli DEIS, Campus

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