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Contents 1 Overview I 2 Overview II 2.1 Example 2.2 Problems: From the Fundamental to the Complex 3 History 4 Non-Linear Dynamics 5 Complex Systems I 6 Complex Systems II 7 Complex Systems III 8 Complex systems IV 8.1 Using computers and algorithms 9 Agent-Based Simulation 10 Agent-Based Simulation: Examples 11 Cellular Automata 12 Open Questions

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

Overview II

Example

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 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...

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

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: where
      1. v_i describes the agent (e.g., its velocity)
      2. f_i gives the non-linear interactions with the other agents
      3. ξ_i 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 adaptive landscape)
   1. equation: where
      1. describes the agent's position
      2. h is a global communication field carrying local information
      3. 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 where
      1. θ_i is the opinion of agent i
      2. w(1 − θ_i | θ_i) rate of opinion change
      3. is the frequency of opposite opinions
      4. κ gives non-linear response to frequency

Open Questions

See Complexity: Open Questions