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Complexity and Simulation

Matthew Roorda

University of Toronto

MAMAMIA – Module 2c

April 23, 2004

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What is a complex system?• One definition:

A complex system is a system for which it is difficult, if not impossible to restrict its description to a limited number of parameters or characterizing variables without losing its essential global functional properties

• More precisely:Complexity deals with non-linear, nested structures, which lead to unexpected higher level behaviours

(Waldrop 1992, cited in Koskenoja and Pas, 2002)

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complex system complicated system

Examples

•  Computer is an example of a complicated system:– The system is composed of many functionally distinct parts

– But the functioning of the system as a whole is (or should be) predictable

• Ecological or economic systems are examples of complex systems– interact non-linearly with their environment

– their components have properties of self-organization which make them non-predictable beyond a certain temporal window

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complex system chaotic system

Complex systems:– Do not reach a stable equilibrium, but neither are they totally

chaotic

– Are systems “at the edge of chaos” where aperiodic systems show “almost periodic” behaviour, even when the evolution path does not repeat itself exactly in a phase diagram

Chaotic systems:– Tiny differences in input quickly become overwhelming differences

in output

– The Butterfly effect – “the notion that a butterfly stirring the air in Peking today can transform storm systems in New York next month”

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Properties of complex systemsProperty One

Non-determinism and non-tractability.

 

Property Two

Limited functional decomposability

 

Property Three

Distributed nature of information and representation

 

Property Four

Emergence and self-organization

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Non-determinism and non-tractability

• Non-determinism: it is impossible to anticipate precisely the behaviour even if we completely know the function of its constituents

• Non-tractability – we can’t fully understand or represent the function of constituent parts of the system anyway!

• Like a fractal – no matter how close you look at it the complexity of the system does not decline.

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No matter how close you look the complexity does not decline

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Limited Functional Decomposability

• a complex system has a dynamic structure

• difficult, if not impossible to study its properties by decomposing it into functionally stable parts

• interaction with the environment and properties of self-organisation allow it to functionally restructure itself

• in other words, the agents themselves learn and/or change their function over time

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Example

• Changes in business self-organization

• Mergers, modes of operation (such as just in time delivery, automation) and ecommerce are changes in self-organization

• These changes are – in response to external changes in technology and economic

conditions, behaviour of competitors

– made so that it can gain a competitive edge over competitors

Firm

Firm

Firm

Firm

Firm

Firm

Merger is a changein self organization

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Distributed nature of information and representation

Two meanings of distributed representation

• Distributed Representation – a system is said to be distributed when its resources (information,

tools, money etc.) are physically or virtually distributed among various individual agents

• Connectionist Model and Robustness- – In the connectionist meaning, a distributed system is one where it is

not possible to localize the resources since they are distributed over multiple actors in a system

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An Example (The household)

• Distributed Representation – within a household, each person plays a different role, keeps track

of different sets of information, and carries out different tasks like child care, etc.

• Connectionist Model and Robustness- – what makes the functioning of a household robust is that

information and functions can pass between household members… I can take over duties that are normally my wife’s responsibility because I know something about those duties

• Many agents in an urban system function with some combination of the distributed representation model and the connectionist model – making them unpredictable and non-deterministic

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Emergence

• Emergence is the process of deriving some new and coherent structures, patterns and properties in a complex system

• Emergent phenomena occur due to the pattern of interactions between the elements of a system over time

 

• Emergent phenomena are observable at a macro-level, even though they are generated by micro-level elements

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A cellular automata demonstration of emergence

The Game of Life– Simple rules -> “emergent behaviour”

http://llk.media.mit.edu/projects/emergence/rules-of-game.html

The Arrow Generator – Different rules -> more complex “emergent behaviour”

http://llk.media.mit.edu/projects/emergence/glider-gun.html

Brian’s Brain – Variations in the initial configuration of the squares can

lead to large changes in the resulting patterns. – But small variations in the underlying rules can lead to

even more dramatic changes

http://llk.media.mit.edu/projects/emergence/mutants.html

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Non-linear systems -> non-predictability

Consider an example of cat and mice populations

Assume that a mouse population is governed by the non-linear equation

 

Xn+1 = kXn – kX2n

 

mouse populationin year n+1

k = “growing factor” (influenced by mouse breeding rate)

decreasing factor

(mice pop can’t grow too much or the cats will eat them)

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Predictability of mouse

population

• As k increases, the system becomes more and more unpredictable

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Bifurcation Diagram for mouse population

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What does this non-linearity example show us?

• Chaotic behaviour can arise even in a very simple system.

• Complexity can arise only from two facts: iteration (feedback from one year to the other) and non linearity in the feedback mechanism

• Even a fully deterministic system can show chaotic behaviour

which means unpredictability over a certain period of time

• Deterministic behaviour can be seen as a special case of chaotic behaviour.

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Interesting Question

• Is our familiar rule based world just an island of intermittency in the midst of chaotic universe?

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Complex systems and Simulation• computer simulations play a central role in complex

systems analysis

• Simulations can be: – outgrowths or natural extensions of the insights of simpler

mathematical models – constructed by modeling directly the (greatly simplified)

features and interactions of the agents in the system being modeled

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Modelling Complexity using Evolutionary Computation

• Cellular Automata– Decentralized, identical components with local connectivity– New state based on the previous state of the cell and its

neighbours– e.g. the Game of Life, TRANSIMS

• Neural Networks– Based on allegory of the brain– setup: each node in the neural net computes a weighted

sum of its input signals from other cells and outputs either a signal or no signal

– training: weights are applied to given inputs to result in the desired outputs

– Meaning behind the weights? Weak behavioural base?

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Modelling Complexity using Evolutionary Computation

• Genetic Algorithms– based on the allegory of the Theory of Evolution

– mainly used as search algorithms

– can be used for parameter estimation in complex systems that are governed by non-linear functions

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Example: Genetic Algorithmsfor non-linear systems

Need to estimate parameters of a mode choice/vehicle allocation model

mode choice/vehicle allocation model is non-linear

maximum likelihood equation is not analytically tractable

use simulation to estimate probabilities

use genetic algorithm to estimate parameters

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Classifier Systems Environment

Agent

Receptors Effectors

Input message list Action message list

If…then rules

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Agent Based Modelling of Complex SystemsA question

• Agent based models -> assume full functional decomposability

• Complex systems -> may have limited functional decomposability

• Agent based models -> assume distributed representation - resources are physically or virtually distributed among agents

• Complex systems -> likely to be a combination of distributed representation and connectionist model

Is the agent based modelling approach limited in its ability to properly model complex systems?

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Question

On the continuum of complexity, where do urban systems lie?

Has implications for the precision/accuracy and ultimately the meaning of the predictions we produce in ILUTE!

Complicated Complex Chaotic

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ResourcesGleick, James. 1987. Chaos: Making a New Science. New York:

Penguin.

Koskenoja, Pia M. and Eric E. Pas. 2002. Complexity and Activity-Based Travel Analysis and Modeling. In In perpetual Motion: Travel Behaviour Research Opportunities and Application Challenges. Mahmassani, H.S. (ed.) New York: Elsevier Science Ltd.

Pavard, Bernard and Julie Dugdale. An introduction to Complexity in Social Science. COSI Project online http://www.irit.fr/COSI/index.php (accessed April 23, 04)

Resnick, Mitchel and Brian Silverman. 1996. Exploring Emergence. Epistemology and Learning Group. MIT Media Laboratory. http://llk.media.mit.edu/projects/emergence/contents.html (accessed April 23, 04)

Sprott’s Fractal Gallery http://sprott.physics.wisc.edu/fractals.htm (accessed April 23, 04)