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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)