ComplexSystems

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Contents

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 \to symmetry
  • Sucess: unification
Mathematical Models
Mathematical Models

Overview II

Example

Example: A Big Chunk of Reality Described by Physics
Example: A Big Chunk of Reality Described by Physics


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?


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


Non-Linear Dynamics


  • A dynamical system is a mathematical model that describes the systems evolution in a state space
  • State variables xi(t) describe the systems dynamics through a set of partial differential equations:
\dot {x}_i(t) = f_i(\vec x, \vec u, t) + \xi_i(t),
where
  • \vec u are the control parameters (external, tunable)
  • fi encodes the non-linear interaction with other states, the control parameters, and the time evolution
  • ξi 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...


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


Complex Systems II


Complex Systems III


Complex systems IV


The New Paradigm
The New Paradigm


Using computers and algorithms

  • New set of tools:
    • agent-based simulations
    • cellular automata


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


Agent-Based Simulation: Examples

  1. Direct communication via agent interaction
    1. equation: \dot {v}_i(t) = f_i(\vec v, \vec \sigma, t) + \xi_i(t), where
      1. vi describes the agent (e.g., its velocity)
      2. fi gives the non-linear interactions with the other agents
      3. ξi is a stochastic term
    2. each agent has one differential equation, i.e., there is no equation for the collective behavior
    3. model for human crowds: [1]
  2. Indirect communication via gradient of field (order parameters \Longrightarrow adaptive landscape)
    1. equation: \dot {\vec r}_i(t) = \frac{\partial h(\vec r, t)}{\partial \vec r} + \vec {\xi}_i(t), where
      1. \vec r_i describes the agent's position
      2. h is a global communication field carrying local information
      3. \vec{\xi}_i is a stochastic term
    2. one equation for each agent and one term $h$ for the collective interaction
    3. the time evolution of $h$ gives insight into the global dynamics of the system
    4. model for ants looking for food


Cellular Automata

  1. Agent-based:
    1. object-oriented
    2. continuously moving agents
  2. Cellular automata
    1. discrete model
    2. features assigned to cells
    3. interaction in local neighborhood
    4. used for modeling social systems
    5. example: voter model w(1-\theta_i | \theta_i) = \kappa(f)f_i^{1-\theta_i} where
      1. θi is the opinion of agent i
      2. w(1 − θi | θi) rate of opinion change
      3. f_i^{1-\theta_i} is the frequency of opposite opinions
      4. κ gives non-linear response to frequency
Cellular Automata Model
Enlarge
Cellular Automata Model

Open Questions

See Complexity: Open Questions