Introductory conceptsBoolean networksComplex networks
An Introduction to Complex Systems Science
Andrea Roli
DEIS, Campus of CesenaAlma Mater Studiorum Universita di Bologna
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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.
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Outline
1 Introductory conceptsComplex systemsMain concepts
2 Boolean networksBasicsRandom Boolean NetworksApplications of Boolean Networks
3 Complex networks
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Outline
1 Introductory conceptsComplex systemsMain concepts
2 Boolean networksBasicsRandom Boolean NetworksApplications of Boolean Networks
3 Complex networks
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
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.
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Complex systems
Examples of complex systems are:
The brainThe societyThe ecosystemThe cellThe ant coloniesThe stock market. . .
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Complex systems science
CSS is interdisciplinary and it involves:
MathematicsPhysicsComputer scienceBiologyEconomyPhilosophy
...just to mention some.
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Complex systems science
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Complex systems science
Some prominent research topics in CSS:
Evolution & emergenceSystems biologyInformation & computationComplex networksPhysics of Complexity
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
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.
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Complex vs. Complicated
Complex: from Latin (cum + plexere); it means “intertwined”.
Complicated: from Latin (cum + plicare); it means “foldedtogether”.
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Properties of complex systems
Complex systems enjoy (some of) these properties:
Composed of many elementsNonlinear interactionsNetwork topologyPositive and negative feedbacksAdaptive and evolvableRobustLevels of organisation
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Outline
1 Introductory conceptsComplex systemsMain concepts
2 Boolean networksBasicsRandom Boolean NetworksApplications of Boolean Networks
3 Complex networks
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Emergence
Emergence refers to understanding how collective propertiesarise from the properties of parts.
A common case of emergence is self-organisation
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Example: Benard cells
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Further examples
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Logistic map: steady states
r ≤ 3→ single value3 < r < 3.57→ repeated sequence of valuesr ≥ 3.57→ sequence of values without apparent structure
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Attractors
AttractorPortion of the state space towards which a dynamical systemevolves over time.
Fixed point(Limit) CycleStrange attractor
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Logistic map: attractors
r ≤ 3→ fixed point3 < r < 3.57→ cycler ≥ 3.57→ strange attractor
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Deterministic chaos
Deterministic modelSensitivity to initial conditionsIn practice, it is impossible to make long term predictionsThe attractor is a strange attractor
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Strange attractor
A strange attractor is a fractalNon-integer dimensionSelf-similarity
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Complexity
Complexity lies at the edge of order and chaos
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex systemsMain concepts
Complexity
Statistical complexity of a systemComplexity = Entropy × Disequilibrium
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Outline
1 Introductory conceptsComplex systemsMain concepts
2 Boolean networksBasicsRandom Boolean NetworksApplications of Boolean Networks
3 Complex networks
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Boolean networks
���������
���������
���������
���������
���������
���������
������������
������������
������������
������������
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Boolean networks
Introduced by Stuart Kauffman in 1969 as a geneticregulatory network model
Discrete-time / discrete-state dynamical system
Non trivial (complex) dynamics
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Structure
Oriented graph of N nodesNode 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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Structure
Oriented graph of N nodesNode i :
- 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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Structure
Oriented graph of N nodesNode 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
X 1
X 2
X 3
AND
OR
OR
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Structure
Oriented graph of N nodesNode 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
X 1
X 2
X 3
AND
OR
OR
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
DynamicsTrajectory in state space
100
011110
101
111
001 010
000
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
DynamicsTrajectory in state space
Trajectory composed of two parts:TransientAttractor
Attractors:Fixed pointsCycles
100
011110
101
111
001 010
000
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
DynamicsTrajectory in state space
Basin of attraction of A:
set of states belonging to the trajectoryending at attractor A
100
011110
101
111
001 010
000
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Dynamics
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Why are BNs interesting?
Minimal complex system
Several important phenomena in genetics can bereproduced
Tight connections with the satisfiability problem
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Outline
1 Introductory conceptsComplex systemsMain concepts
2 Boolean networksBasicsRandom Boolean NetworksApplications of Boolean Networks
3 Complex networks
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean 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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Random Boolean networksProperties
K = 1: ORDER
Frozen dynamicsExtremely robust: small perturbations die out quickly
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Random Boolean networksProperties
K ≥ 3: (pseudo) CHAOS
Very long cycles (∼ 2N )Sensitivity to initial conditionsNot robust: small perturbations spread quickly throughoutthe system
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Random Boolean networksProperties
K = 2: CRITICALITY
Short cycles (∼ low degree polynomial of N)Robust: small perturbations die out (in the long term) orkeep smallSecond order phase transition
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Critical parameters
From the theory:
Kc = [2pc(1− pc)]−1
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Extensions and variants
AsynchronousProbabilisticMultivalued logicsContinuous variables ruled by differential equations (e.g.,Glass networks)Multiple interacting BNs
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Outline
1 Introductory conceptsComplex systemsMain concepts
2 Boolean networksBasicsRandom Boolean NetworksApplications of Boolean Networks
3 Complex networks
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
BNs in biology
Cellular dynamics models
Models of specific genetic regulatory networks
Cancer and stem cell models
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Cell dynamics
Main resultAttractor (or set-of)↔ cell type
Cancer and stem cell models
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Cell dynamics
Main resultAttractor (or set-of)↔ cell type
Cancer and stem cell models
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Robustness & adaptiveness
Main resultCritical BNs make it possible to achieve the best balancebetween robustness and adaptiveness.
Real cells are critical
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
Robustness & adaptiveness
Main resultCritical BNs make it possible to achieve the best balancebetween robustness and adaptiveness.
Real cells are critical
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
KO avalanches
Main resultBNs can reproduce the same avalanche distribution as realgenetic networks.
Same results with several kinds of BNs (e.g., Glass nets)
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
KO avalanches
Main resultBNs can reproduce the same avalanche distribution as realgenetic networks.
Same results with several kinds of BNs (e.g., Glass nets)
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
BNs in engineering and computer science
Satisfiability problem
Learning systems
Boolean network robotics
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
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.
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
BasicsRandom Boolean NetworksApplications of Boolean Networks
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.
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex networks
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Complex networks
System’s behaviour depends on the structure of relationamong the components
Useful models from graph theory
Recent research stream in CSS
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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.
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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)
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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)
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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)
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Random graphs
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Features of interest
Vertex degree (in- and out-degree if edges are oriented)
Diameter, characteristic path length et similia
Clustering coefficient
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Characteristic length L(G)
Informally: average path length between any pair of vertices.
Random graphs→ short L(G)
Grid graphs→ long L(G)
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Characteristic length L(G)
Informally: average path length between any pair of vertices.
Random graphs→ short L(G)
Grid graphs→ long L(G)
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Scale-free nets
Scale-free networks can represent the topology of:
Social relations (e.g.,scientific collaborations)Web-pagesThe Internet. . .
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Scale-free nets
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
Scale-free nets
Dynamics dramatically different from random and regulartopologiesImplications in medicine (e.g., epidemics), society, InternetRelated to small-world phenomena (low length, highclustering)
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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
Andrea Roli An Introduction to Complex Systems Science
Introductory conceptsBoolean networksComplex networks
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)
Andrea Roli An Introduction to Complex Systems Science