Complex Systems Tutorial

Jan Burian
honzaburian (at)

You can find here:


Intuitive definitions of complexity

Basics of (complex) systems science

Self-organization and related concepts

Formal definitions of complexity

Very short introduction to modeling methodology

Cellular automatons

Complex networks

Agent-based models


Intuitive Definitions of Complex Systems

System is an entity in terms of parts and relations between them.

Complex system (complex comes from Latin com-  together + plectere  to twine or braid) is a system composed from relatively many mutually related  parts.

Complex systems are usually (but not always) intricated - hard to describe or understand.

Examples of complex systems
Parts of human society:
  • Markets
  • Organizations
  • Language
  • Internet
  • Cells
  • Organ – e.g. brain
  • Immune system
  • Organisms
  • Populations
  • Ecosystem
  • Turbulence
  • Weather
  • Percolation
  • Sandpile

The world consists of many complex systems, ranging from our own bodies to ecosystems to economic systems. Despite their diversity, complex systems have many structural and functional features in common that can be effectively simulated using powerful, user-friendly software. As a result, virtually anyone can explore the nature of complex systems and their dynamical behavior under a range of assumptions and conditions. (M. Ruth, B Hannon, Dynamic Modeling Series Preface)

Structurally complex system
A system that can be analyzed into many components having relatively many relations among them, so that the behavior of each component can depend on the behavior of many others. (Herbert Simon)

Remember: The number of relationships could be much higher than the number of components!

Dynamically complex system
A system that involves numerous interacting agents whose aggregate behaviors are to be understood. Such aggregate activity is nonlinear, hence it cannot simply be derived from summation of individual components behavior. (Jerome Singer)

System types  Simple structure Complex structure
Simple dynamics Example: Pendulum
Model: Analytical - differential equations

Example: Closed reservoir of gas
Model: Statistical equations

Complex dynamics

Example: Double pendulum
Model: Analytical - Complex differential equations
or simple simulations

Phase portrait of Double pendulum

Example: Ant pile
Model: Multi-agent models



Basics of (complex) systems science

Remember: Interconnection of parts matters in complex systems!

Five blind man try do describe an elephant - to fully understand the system we must view it from different points of view.

Two complementary approaches to system behaviour

Reductionism: The properites of the whole system could be explained in terms of its parts. Holism : The whole system cannot be determined or explained by its component parts alone. Instead, the system as a whole determines in an important way how the parts behave.
  • Understanding of the parts leads to understanding of the whole
  • Accent on parts
  • To understand the whole we must understand also the relations between the parts in the whole system
  • Accent on relationships

Pragmatic approach rests on a combination of both reductionism and holism.


The basic rules of the complex systems could be paradoxically very simple, but their effects are intricate and unexpected because of feed-back relations between parts.

The return of a portion of the output of a process or system to the input.

System dynamics

State space (phase space) is an abstract space in which all possible states of a system are represented, with each possible state of the system corresponding to one unique point in the state space. Dimensions of state space represent all relevant parameters of the system. For example state space of mechanical systems has six dimensions and consists of all possible values of position and momentum variables.

Dynamics of the system is the set of functions that encode the movement of the system from one point in the state space to another.

Trajectory of the system is the sequence of system states.

Fixed point is a point in the state space where the system is in equilibrium and does'nt change.

Attractor is a part of the state space where some trajectories end.

Attractor types

Fixed point

Limit cycle

Strange attractor

Dynamical systems can often be modeled by differential equations dx/dt=v(x), where x(t)=(x1(t), …, xn(t)) is a vector of state variables, t is time, and v(x)=(v1(x), …, vn(x)) is a vector of functions that encode the dynamics. For example, in a chemical reaction, the state variables represent concentrations. The differential equations represent the kinetic rate laws, which usually involve nonlinear functions of the concentrations. Such nonlinear equations are typically impossible to solve analytically, but one can gain qualitative insight by imagining an abstract n-dimensional state space with axes x1, …, xn. As the system evolves, x(t) flows through state space, guided by the ‘velocity’ field dx/dt = v(x) like a speck carried along in a steady, viscous fluid. Suppose x(t) eventually comes to rest at some point x*. Then the velocity must be zero there, so we call x* a fixed point. It corresponds to an equilibrium state of the physical system being modeled. If all small disturbances away from x* damp out, x* is called a stable fixed point — it acts as an attractor for states in its vicinity. Another long-term possibility is that x(t) flows towards a closed loop and eventually circulates around it forever. Such a loop is called a limit cycle. It represents a self-sustained oscillation of the physical system. A third possibility is that x(t) might settle onto a strange attractor, a set of states on which it wanders forever, never stopping or repeating. Such erratic, aperiodic motion is considered chaotic if two nearby states flow away from each other exponentially fast. Long-term prediction is impossible in a real chaotic system because of this exponential amplification of small uncertainties or measurement errors. (Strogatz, 2001)

Example: The predator-prey model

We study the dynamics of mutually dependent population size of predators and prey.

The model is supported by analyses of 100 year fur trapping records of the Hudson's Bay Company.


Animation of the system dynamics and the two differential equations governing the dynamics (Lotka-Volterra equations). X represents the size of hare population and Y the size of lynx population.

The portrait of dynamics for different initial x and y parameters forms a fractal.
More information about this fractal.

Formal definitions of complexity

Non suitable complexity measures for complex systems

There are many formal definitions of complexity available. Only a small portion of them is suitable for description of complex systems. There are two particular notions of complexity which are not suitable for description of complex systems but have very good sense in other domains.

Suitable complexity measures

Good measures of system complexity should measure the amount of regularities in the system (and not its randomness). Such measures should be low for both very simple systems (where is only one or very few dominant regularities) and random systems (where are almost no regularities). Such measures are called non-monotonic. We can say that they are somewhere between order and randomness - on the „edge of chaos“ (Langton, 1990).


Neural Complexity

Neural complexity (Sporns et al., 2000, 2002) is a measure inspired by the cognitive processes in the brain. It measures how much the change of activity in one part of he network changes the activity in other parts. The authors described it shortly as a measure of "the difference that makes difference". Neural complexity is one of many complexity measures based on mutual information.

Mutual information between two parts of a system is defined:

MI(Xjk,, X-Xjk) = H(Xjk) + H(X- Xjk) – H(X)

There X is the system, Xjk is the j-th permutation of a part of size k and X-Xjk is the rest of the system.

Neural complexity is formally the sum of average mutual information between subsets of the system and the rest of the system

The neural anatomy, neural activity, EEG signal and neural complexity of the brain of an old (a), adult (b) and very young (c) cat. In the old cat there are mostly local specialized connections present but the global integrative connections are missing due to degenrative processes, the result is rather random dynamics. In the adult cat there are both local and global connections present, the result is complex dynamics. In the young cat the local connections are not developed yet but the global connections are already present, the result is regular dynamics (Edelman & Tononi, 2000).

Statistical Complexity

The statistical complexity (Shallizi 2001, 2003, 2004) reflects the intrinsic difficulty of predicting the future states of the system from the system history. It is the  amount of information needed about the past of a given point in order to optimally predict its future. Systems with a high degree of local statistical complexity are ones with intricate spatio-temporal organization. Statistical complexity is low both for highly disordered and trivially-ordered systems.

Self-organization and related concepts


Self-organization (First used by Ashby in 1948.). The ability of the system to autonomously (without being guided or managed by an outside source)  increase its complexity.

If a local system is an open system receiving relatively stable and appropriate amount of energy from its environment and the local system is composed from sufficient number of parts which are able to interact through positive and negative feedback, there could (depending on some parameters) be established relatively stable network of feedback loops. This process is called self-organization and the established dynamic network is called self-organized system.

Examples of self-organisation:

In Physics:

Benard cells - coherent motion of large number of molecules in heated liquid layers.

Belousov-Zhabotinsky reactions - a specific "coctail" of chemical ingredients loops through visualy discernable states when it receives thermal (heated) or mechanical (stirred) energy.

In Biology:

Flocking - a group of organism can self-organise in a relatively coherent whole which is able to synchronously react on external stimuli.

Ants demo - increasing the length of pheromone trace or the number of ants would lead to self-organization of the food track (result of stigmergic interaction between agents and environmnent).

An ecosystem or a whole biosphere - the feedback loops between environment and organism could lead to stabilisation (homeostatis) of some environmental parameters.

Emergence and self-organization

Traditional definition of emergence
The arising of characteristics of the whole which cannot be attributed to the parts. There arise new qualitative and not only quantitative changes. Very vaguely: the whole is more than sum of it parts (an statement made already by Aristotle in Metaphysics).

Modern definition
Emergence is „the arising of novel and coherent structures, patterns and properties during the process of self-organization in complex systems." (Corning, 2002; Goldstein, 1999)

Common characteristics of emergence:
  1. Radical novelty (features not previously observed in the system)
  2. A global or macro “level” (i.e., there is some property of “wholeness”)
  3. Coherence or correlation (integrated wholes that maintain themselves over some period of time)
  4. It is the product of a dynamical process (it evolves)
  5. It is meaningfull for us (i.e. has some pragmatic value for us – we can use it).

Weak emergence: new properties arising in systems as a result of the interactions at an elemental level. The causal conection between the interactions of the parts and the properties of the whole can be traced in great detail.

Strong emergence: the properties of the whole supervene on the properties of the parts. Supervenience describes  causal dependence between sets of properties. If property B is causaly dependent on property A, it means that one state of property B could be caused by many states of property A, but one state of property A causes exactly one state of property B.



The combined (cooperative) effects that are produced by two or more particles, elements, parts or organisms – effects that are not otherwise attainable. (Corning, 2002)

Example: Lichen and other symbiotic organisms


Adaptability is the ability of a system to maintain its complexity in changing environment. Often we can find a feed-back between system and its environment.

System types Constructed Self-organized
Non-adaptable Example: Classical machines Example: Crystals
Adaptable Example: Adaptable robots Example: Living organisms

Daisy world – an example of self-organized adaptive system

NetLogo Daisy world model
Generalized model at NANIA

Cellular automatons (CA)



1D cellular automatons

Basic description of a simple cellular automaton (CA) as presented in (Wolfram, 2002).
This book is available on internet and represents a good introduction to cellular automatons (and other simple automatons with complex behavior) but it also gained bad reputation due its egocentric tone.

Basic types of rules

Different rules produce very different behavior.

Four classes of behavior in 1D CA (Wolfram, 1983)

2D Cellular automatons and the Game of Life

Types of 2D neighborhood
von Neumann

Game of Life

From the five cells on the left (so called F-pentomino) evolved one hundred steps a complex pattern.

Left: An example of an complex oscilator in Life (Gosper's glider gun). Right: Turing machine implemented in Life (Rendell, 2005).

More about CA dynamics

Attractor basins structure and entropy variation in different classes of CA rules (Wuensche, 1998)


Self-replication was first investigated by von Neumann in 1940s. The von Neumann self-reproducing automata is actually a universal constructor' that constructs "any machine'' in its 29-state cellular space. In particular, it is capable of Turing universal computation. It solves the self-reproduction problem by reading a tape containing instructions on how to build a copy of itself, provides the copy with a copy of its own input tape, and then presses the ON button starting the copy in operation. In the 1980s, C. Langton and then J. Byl showed that in fact much smaller automata can in fact self-reproduce.

Langton's self-replicating loop

Computational universality of some CAs

CA rule 110


Generalizations of CAs

Applications of CA

Resources about CA on WWW

Mireks's Java Cellebration - one of the best CA tools

Game of Life and Cellular automaton on Wikipedia

CA Tutorial by Alexander Schatten,

"History of Cellular Automata" from Stephen Wolfram's "A New Kind of Science"

General article about Cellular Automata by Cosma Shallizi

The DDLab manual by Andy Wuensche with many information about CA, discrete dynamical networks and their attractor basins.

Complex Networks

In CA there was interaction possible only between neighbouring cells in a spatial matrix. But the interaction between active parts of a system could be generally described by a network where the active components are represented by nodes and the interactions by edges. Complex networks are a subgroup of networks with "interesting" properties. Natural and social networks are often complex.

Motivation for the study of complex networks:

Examples of complex networks
  • Human society
    • Social networks
      • Economics
      • Epidemiology
      • Collaboration and Citation networks
      • Spreading of innovations
    • Electrical grid
    • Internet


The 1318 transnational corporations that form the core of the economy. (Vitali S. et al., 2011)

Structure of internet (nodes - servers, edges - connections)
(Hal Burch and Bill Cheswick, Lumeta Corp.)

  • Food chains
  • Gene regulation networks
  • Metabolism networks
  • Neural networks

Structure of yeast protein interactions (nodes - proteins, edges - reactions) (Barabasi et. al., 2003)

Macaque cortex network (Young, 1993)

For more examples see Gallery of network images

Basic notions of Graph theory

Some properties of complex networks:

Types of complex networks

Types of complex networks (Huang, 2005)

Random network

Small World network

Scale-free network

Random networks

Small world networks

Scale-free networks

Comparison between the degree distribution of scale-free networks (circle) and random graphs (square) having the same number of nodes and edges. For clarity the same two distributions are plotted both on a linear (left) and logarithmic (right) scale. The bell-shaped degree distribution of random graphs peaks at the average degree and decreases fast for both smaller and larger degrees, indicating that these graphs are statistically homogeneous. By contrast, the degree distribution of the scale-free network follows the power law, which appears as a straight line on a logarithmic plot. The continuously decreasing degree distribution indicates that low-degree nodes have the highest frequencies; however, there is a broad degree range with non-zero abundance of very highly connected nodes (hubs) as well. Note that the nodes in a scale-free network do not fall into two separable classes corresponding to low-degree nodes and hubs, but every degree between these two limits appears with a frequency given by P(k). (Albert, 2005)

The Potential Implications of Scale-Free Networks (Barabasi et al., 2005):

Resources about complex networks

Chapter about networks in Complex Science for a Complex World.

Exploring complex networks, an article by Strogatz in Nature.

Scale-free networks, an article by Barabasi and Bonabeau in Scientific American.

NetLogo models:

Very short introduction to modeling methodology

Models are always “wrong” but sometimes could be useful! (Georg E. P. Box)

A different look at logical relationship between a multiagent model and reality:
Axelrod (2003) points out: “like deduction model starts with a set of explicit assumptions. But unlike deduction, it does not prove theorems. Instead, a simulation generates data that can be analyzed inductively”. Induction comes at the moment of explaining the behavior of the model. It should be noted that although induction is used to obtain knowledge about the behavior of a given model, the use of a model to obtain knowledge about the behavior of the real world refers to the logical process of abduction. Abduction, also called inference to the best explanation, is amethod of reasoning in which one looks for the hypothesis that would best explain the relevant evidence, as in the case when the observation that the grass is wet allows one to suppose that it rained. (Encyclopedia of Complexity)

Steps of modeling:

Agent-based models (ABM)


The term "agent" means an active, autonomous and situated unit.

Relation between ABM and Multi-agent systems (MAS)

ABM are a subclass of Multi-agent systems (MAS). Typically in MAS agents could be biological or artificial entitities situated in a real world like a group of animals, group of cooperating robots or virtual entities situated in a non-simulated environment like software agents acting in a computer network. In ABM agents are typically software objects (inter) acting in a simulated environment. ABM could be interpreted as models of real-world MAS.

Characteristics of ABM:

When to use ABM:

Which features of real complex systems can we better understand with the help of ABM?

Logic and modeling

In every model there are present aspects of deduction, induction and abduction. But according to the questions we ask the emphasis could be on different types of logical reasoning.

Example: Evolution of cooperation in iterated prisoners dilemma models

Prisoner's dilemma in game theory
Two agents decide between cooperation and non-cooperation and are rewarded after their decisions.

Three basic situations could arise:

Iterated prisonner's dilemma (IPD)
The decisions of agents are repeated and rewards accumulated.

The tournament of different playing strategies surprisingly showed that the best strategy for IPD is very simple:
Tit for Tat: If you cooperate I also cooperate, if you don't cooperate I also don't cooperate.

If we encode the strategies into a simple genome and evolve competing populations of these strategies the Tit for Tat strategy will evolve spontaneously and become dominant for a reasonable range of rewards.

Other examples:


Agent-based computational economics (ACE)

(Graphic by T. Eymann)

Aims of ACE (according to Tesfatsion):

ABM and micro-economical models share the bottom-up approach but in other aspects they substantially differ.

In the process of formalizing a theory into mathematics it is often the case that one or more — usually many! — assumptions are made for purposes of simplification; representative agents are introduced, or a single price vector is assumed to obtain in the entire economy, or preferences are considered fixed, or the payoff structure is exactly symmetrical, or common knowledge is postulated to exist, and so on. It is rarely desirable to introduce such assumptions, since they are not realistic and their effects on the results are unknown a priori, but it is expedient to do so. ... it is typically a relatively easy matter to relax such ‘heroic’ assumptions-of-simplification in agent-based computational models: agents can be made diverse and heterogeneous prices can emerge, payoffs may be noisy and all information can be local.(Axtel, 2000)

Micro-economical models
analytical solutions
computational synthesis
looking for equilibrium
dynamical systems often without any equilibrium
description of behavior
emergent behavior
homogenous agents
non-homogenous agents
based on variables
based on relations

For comprehensive overview see ACE web pages by Leigh Tesfatsion.

Two particularly interesting models implemented in NetLogo:

Artificial stock market by Carlos Goncalves

Model of market without intermediation by Michal Kvasnička

Software tools for ABM




Examples of extremely simple NetLogo Forrest fire model and its modification for beginners.




For other ABM tools and more detailed comparation see

Resources about ABM

A guide for newcomers to agent based modeling in the social sciences - by Robert Axelrod and Leigh Tesfatsion

ACE web pages by Leigh Tesfatsion

Agent-based modeling: Methods and techniques for simulating human systems by Eric Bonabeau

Seeing around corners - a popular article about ABM by Jonathan Rauch

Why agents? On the varied motivations for agent computing in the social sciences - an elaborated analysis of relations between ABM and classical analytical models, by Robert Axtell

From factors to actors: Computational sociology and agent-based modeling - sociological approach to ABM by Michael W. Macy and Robert Willer

Complexity of Cooperation Web - by Robert Axelrod

Twenty Years on: The Evolution of Cooperation Revisited - an overview by Robert Hoffmann

Parochial altruism resources


Albert, R. J Cell Sci 2005;118:4947-4957

Axelrod, R. (2003): Advancing the Art of Simulation in the Social Sciences. Japanese Journal for Management Information System, Special Issue on Agent-Based Modeling, Vol. 12, No. 3, Dec. 2003.

Axelrod, R. (1987): The Evolution of Strategies in the Iterated Prisoner's Dilemma. In Lawrence Davis (ed.),Genetic Algorithms and Simulated Annealing. London: Pitman, and Los Altos, pp. 32-41. available on WWW: <>.

Barabasi, A. L. Bonabeau, E. Scientific American. New York: May 2003. Vol. 288, Iss. 5; p. 60.

Cools, S.B., C. Gershenson, and B. D'Hooghe (2006): Self-organizing traffic lights: A realistic simulation. available on WWW: <>.

Corning, Peter A. (2002), The Re-Emergence of "Emergence: A Venerable Concept in Search of a Theory. Complexity 7(6): 18-30

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