# 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

# Overview II

### Example

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?"
• 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

# 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: $\dot {v}_i(t) = f_i(\vec v, \vec \sigma, t) + \xi_i(t),$ 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