ComplexSystems
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Contents |
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Overview I
Fundamental
- Idea: science is the quest to capture the processes of nature in formal mathematical representations
- Paradigm: mathematical models of reality are independent of their formal representation symmetry
- Sucess: unification
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Overview II
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Example
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Problems: From the Fundamental to the Complex
- Real systems have:
- a multitude of interacting sub-parts
- non-linear and chaotic effects
- no general closed-form analytical solutions
- How to get from quarks to the human mind?
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History
The paradigm shift to systems thinking: interdisciplinary field which studies systems as a whole, focusing on the relationships of the system elements.
The World as a System
- General Systems theory: unification of systems; motivated from biology; 1940s
- Cybernetics: theory of communication and control; functional relationship of parts; regulatory feedback; 1950s
- Dissipative structures: thermodynamic systems far from the equilibrium state; 1970s
- Synergetics: self-organization in open systems; external control parameters; order parameters; 1970s
- Catastrophe theory: small changes in parameters lead to big changes in systems dynamics; 1970s
- Chaos theory: non-linear effects; self-similarity; initial conditions; path dependence; 1980s
- The "new" theory of complexity: complex adaptive systems; self-organization; emergence; multi-agent simulations; 1990s
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Non-Linear Dynamics
- A dynamical system is a mathematical model that describes the systems evolution in a state space
- State variables x_{i}(t) describe the systems dynamics through a set of partial differential equations:
- where
- are the control parameters (external, tunable)
- f_{i} encodes the non-linear interaction with other states, the control parameters, and the time evolution
- ξ_{i} is the time-dependent noise (stochastic term)
- Summary:
- vastly different behavior from the same dynamical system: ordered, complex, chaotic
- instability controlled by control parameter
- instability controlled by feedback mechanism
- simple systems with complex dynamics
- however, still at the macro view...
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Complex Systems I
- Shift from the macro to the micro view
- Challenge: "how does the macro behavior emerge from the interaction of the system elements?"
- New paradigms:
- shift from analytical to algorithmic approach
- simple rules lead to complex behavior
- Definition: complex systems are systems with multiply interacting components whose behavior cannot be inferred from the behavior of the components
- Buzzwords:
- self-organization
- emergence
- adaptivity
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Complex Systems II
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Complex Systems III
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Complex systems IV
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Using computers and algorithms
- New set of tools:
- agent-based simulations
- cellular automata
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Agent-Based Simulation
- Ideas: agents follow local rules and generate global structures (emergence).
- Elements:
- non-linear feedback/coupling of agents
- collective generation of order parameters
- order parameters restrict agents
- competition and selection during establishment of order parameters
- stochastic fluctuations included
- interaction as communication
- Characteristics:
- no over-arching strategy
- path-dependence; the system has a unique history
- spontaneous emergence of order
- instability as a key element
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Agent-Based Simulation: Examples
- Direct communication via agent interaction
- equation: where
- v_{i} describes the agent (e.g., its velocity)
- f_{i} gives the non-linear interactions with the other agents
- ξ_{i} is a stochastic term
- each agent has one differential equation, i.e., there is no equation for the collective behavior
- model for human crowds: [1]
- equation: where
- Indirect communication via gradient of field (order parameters adaptive landscape)
- equation: where
- describes the agent's position
- h is a global communication field carrying local information
- is a stochastic term
- one equation for each agent and one term $h$ for the collective interaction
- the time evolution of $h$ gives insight into the global dynamics of the system
- model for ants looking for food
- equation: where
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Cellular Automata
- Agent-based:
- object-oriented
- continuously moving agents
- Cellular automata
- discrete model
- features assigned to cells
- interaction in local neighborhood
- used for modeling social systems
- example: voter model where
- θ_{i} is the opinion of agent i
- w(1 − θ_{i} | θ_{i}) rate of opinion change
- is the frequency of opposite opinions
- κ gives non-linear response to frequency
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