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Complexity and Simulation

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  1. Complexity and Simulation Matthew Roorda University of Toronto MAMAMIA – Module 2c April 23, 2004

  2. 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)

  3. 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

  4. 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”

  5. Properties of complex systems Property 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

  6. 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.

  7. No matter how close you look the complexity does not decline

  8. 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

  9. 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 Merger is a changein self organization Firm Firm

  10. 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 possibleto localizethe resources since they are distributed over multiple actors in a system

  11. 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

  12. 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

  13. 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

  14. 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 k = “growing factor” (influenced by mouse breeding rate) decreasing factor (mice pop can’t grow too much or the cats will eat them) mouse populationin year n+1

  15. Predictability of mouse population • As k increases, the system becomes more and more unpredictable

  16. Bifurcation Diagram for mouse population

  17. 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.

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

  19. 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

  20. 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?

  21. 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

  22. 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

  23. Classifier Systems Environment Agent If…then rules Input message list Action message list Receptors Effectors

  24. 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?

  25. 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

  26. Resources Gleick, 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)