An introduction to complex systems science
Andrea Roli
Dipartimento di Informatica–Scienza e Ingegneria (DISI) Campus of Cesena
Alma Mater StudiorumUniversit `a di Bologna [email protected]
Disclaimer
The field of Complex systems science is wide and it involves many subjects and disciplines.
These slides just provide an informal introduction to some relevant topics in this area.
Complex systems
Examples of complex systems are: The brain
The society The ecosystem The cell
The ant colonies The stock market
Complex systems science
A new interdisciplinary field of science studying how parts of a system give rise to the collective behaviours of the system, and how the system interacts with its environment. It focuses on certain questions about parts, wholes and relationships.
Complex systems science
Three main interrelated approaches to the modern study of complex systems:
1 Understanding the ways of describing complex systems
2 Understanding the process of formation of complex
systems through pattern formation and evolution
Complex systems science
Three main objectives:
1 Understand
2 Make predictions
Complex systems science
Some prominent research topics in CSS: Evolution & emergence
Systems biology
Information & computation Complex networks
Physics of Complexity
Reductionism vs. Holism
Reductionism: an approach to understanding the nature of complex things by reducing them to the description of their parts.
Holism: idea that the properties of a system cannot be
determined or explained by its component parts alone, but they are rather explained in terms of the interactions among the parts. Summarised with the sentence “The whole is more than the sum of its parts”.
Complex vs. Complicated
Complex: from Latin (cum + plexere); it means “intertwined”.
Complicated: from Latin (cum + plicare); it means “folded together”.
Properties of complex systems
Complex systems are characterised by (some of) these properties:
Composed of many elements Nonlinear interactions
Network topology
Positive and negative feedbacks Adaptive and evolvable
Robust
Levels of organisation (tangled hierarchies) Emergent phenomena
Emergence
“Emergence refers to the arising of novel and coherent structures, patterns, and properties during the process of self-organization in complex systems. Emergent phenomena are conceptualized as occurring on the macro level, in contrast to the micro-level components and processes out of which they
arise.”(from J. Goldstein, Emergence as a Construct,
Emergence1(1):49–72)
“Emergence refers to all the properties that we assign to a system that are really properties of the relationship between a
system and its environment.”(from Y. Bar-Yam, Concepts in
Self-organisation
A prominent case of emergenceDynamical mechanisms whereby structures appear at the global level from interactions among lower-level components.
Creation of spatio-temporal structures
Possible coexistence of several stable states (multistability) Existence of bifurcations when some parameters are varied
Further examples of emergence
Synchronisation in fireflies
Foraging and nest building in insects Flocking
Organs and tissues in multicellular organisms∗
Clouds∗
Factions in political parties∗
Universality
CSS tries to identify principles that can beuniversally
applied to some system classes
In spite of specific differences, many systems exhibit the same properties
Model of a system
A model is an abstract and schematic representation of a
system. It is also usually aformalrepresentation of the system.
It makes it possible to:
1 investigate some properties of the system
2 make predictions on the future
It is usually in the form of a set of objects and the relations among them.
Properties of a model
It represents a portion of the system
It only captures some of the system’s features The abstraction process involves simplification, aggregation and omission of details
Example: the logistic map
xt+1=rxt(1−xt)
xi ∈[0,1]
r ∈[0,4]
Simple model of population growth
Logistic map: steady states
r <3→single value
3≤r <r∞≈3.57→repeated sequence of values
r∞≤r ≤4→sequence of values without apparent
Attractors
Attractor: Portion of the state space towards which a dynamical system evolves over time.
Fixed point (Limit) Cycle
Logistic map: attractors
Deterministic chaos
Deterministic model
Sensitivity to initial conditions
In practice, it is impossible to make long term predictions
Strange attractor
A strange attractor is afractal
It has structure at arbitrarily small scales Self-similarity
Chaos and fractals in the real world
Heart beat
Growth phenomena Urban development Stock market
Complexity
Boolean networks
Introduced by Stuart Kauffman in 1969 as a genetic regulatory network model
Discrete-time / discrete-state dynamical system
X1
X2
X3
AND
Dynamics
System state at timet:s(t) = (x1(t), . . . ,xN(t)) Dynamics controls node update
Synchronous vs. asynchronous dynamics
Synchronous dynamics (and deterministic update rules): One successor per state
Dynamics
Transition function t t+1 x1 x2 x3 x1 x2 x3 0 0 0 0 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 1 1 1 1 1 1 0 0 0 1 1 1 0 1 0 1 1 1 1 0 0 1 1 1 1 1 1 1 1 X1 X2 X3 AND OR ORDynamics
Trajectory in state space
Trajectory composed of: Transient Attractor Attractors: Fixed points Cycles 100 011 110 101 111 001 010 000
Why are BNs interesting?
“Minimal” complex system
Important phenomena in genetics can be reproduced
KO gene expression avalanches Cell differentiation
Ensemble approach for supporting the “living systems are critical” conjecture
Tight connections with the satisfiability problem
Random Boolean networks
ModelK inputs per node
Inputs chosen at random, no self-arcs
→outgoing arcs Poissonian distributed
Random Boolean functions: each entry of truth table has
Random Boolean networks
PropertiesK =1: ORDER
Frozen dynamics
Extremely robust: small perturbations die out quickly
Random Boolean networks
PropertiesK ≥3: CHAOS
Very long cycles (∼2N)
Sensitivity to initial conditions
Not robust: small perturbations spread quickly throughout the system
Random Boolean networks
PropertiesK =2: CRITICALITY
Short cycles (∼low degree polynomial ofN)
Robust: small perturbations die out (in the long term) or keep small
Critical parameters
From the theory:
Properties at the edge of order and chaos
Large fluctuations Phase transitions Criticality Long-range correlations Percolation Scale-freePhase transitions
Depending on somecontrol parametersthe system can
display different kinds of behaviours
The phase transition occurs at acriticalvalue of the
The Ising model
Well known model in statistical physics Simple description, complex behaviour
The forest-fire model
Lattice in which cells can be either occupied by a tree or empty
Cells are initially assigned by means of a Bernoulli
distribution of parameterp
Forp<pc ≈0.59, the fire dies quickly; ifp>pcthe fire spans the whole area very fast
Atp=pc there is a phase transition⇒percolation(i.e.
Self-organised criticality
There exist systems that tend to organise their dynamics such
that thecritical stateis maintained
Earthquakes
Avalanches (snow, neuronal, etc.) Forest Fires
Biological evolution
The importance of being critical
Evidence supporting the conjecture that living systems are critical. E.g. cells and the brain
Some systems maximise their computational power in the critical region
Critical systems seem to achieve an optimal trade-off between robustness and evolvability
At the critical region, information flow is maximised Maximal computational capabilities are attained at criticality (e.g. spiking neurons, reservoir computing models)
Complex networks
System’s behaviour depends on the structure of relation among 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.
Random graphs
First model of the topology of a complex system Interesting theoretical results
Baseline for comparison with other topologies
Strictly speaking, arandom graph modelis defined in terms of
an 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)
Scale-free nets
Scale-free networks can represent the topology of: Social relations (e.g.,scientific collaborations) Web-pages
The Internet
Scale-free nets
Degree distribution:
number of vertices with degreek ∼k−γ
Few vertices with many connections (hubs) and many
vertices with few connections Robust against accidental damages Fragile w.r.t. specific attacks
Scale-free nets
Dynamics dramatically different from random and regular topologies
Implications in medicine (e.g., epidemics), society, Internet
Often related tosmall-world phenomena (low path length,
Scale-free nets development
A net evolves to a scale-free topology if the following two conditions hold (sufficient condition):
Growth: older vertices have a higher number of connections
Preferential attachment: new vertices tend to be attached to vertices with many connections (prob. is proportional to the number of links)
Information theory for complex systems
science
Information-theoretic measures in CSS
A computational process can be seen as the evolution in time of a dynamical system
Information theory can be applied to quantify some properties of the system
Used in the definition of measures of complexity Currently used in biology, neuroscience, robotics and complex systems data analysis in general
Commonly used measures in CSS
Entropy: H(X) =− P x∈X P(x)logP(x) Mutual information:I(X;Y) =H(X) +H(Y)−H(X,Y) Disequilibrium:D(X) = P x∈X P(x)− 1 |X | 2Statistical complexity
Measures the algorithmic complexity of a program that reproduces the statistical properties of a system (notion proposed by Crutchfield)
Statistical complexity of the Ising model
0.02 0.04 0.06 a ver age LMC comple xityOther measures of complexity
-machine complexity: measures the statistical complexity of a reconstructed model of the system
Set-based complexity: measures the complexity of an ensemble of pieces of information (e.g. genes in a cell)
Neural complexity: measures the complexity of a system w.r.t. the complexity of the environment
Additional material
“An introduction to complex systems science” lecture notes,
http://campus.unibo.it/183308/1/intro-css v4.pdf
Complexity explorer,www.complexityexplorer.org
Selected bibliography (1)
Bar–Yam, Y.
Dynamics of Complex Systems.
Studies in nonlinearity, Addison–Wesley, Reading, MA (1997)
S.A. Kauffman.
The Origins of Order: Self-Organization and Selection in Evolution.
Oxford University Press, 1993.
R. Serra and G. Zanarini.
Complex Systems and Cognitive Processes.
Selected bibliography (2)
R. Sol ´e and B. Goodwin.
Signs of life.
Basic Books, 2000.
S.H. Strogatz.
Nonlinear dynamics and chaos.
