# On the complexity of a social behavior model

C. Somarakis, J. S. Baras

Proceedings of the Eighth International Conference on Complex Systems (ICCS 2011), pp. 1330-1344, Boston Marriott, Quincy, MA, USA, June 26 - July 1, 2011.

Abstract

Cellular automata are perhaps the simplest mathematical representations of complex dynamical systems and networks. They are spatially and temporally discrete, deterministic models characterized by local interaction and an inherently parallel form of evolution.
The history of cellular automata can be traced back to 1948, when J. L. von Neumann introduced them to study simple biological systems[2]. The widespread popularization of these systems was achieved in the 1980s through the work of S. Wolfram who gave an extensive classification of cellular automata as mathematical models for self-organizing statistical systems . Wolfram's systematic research is to relate cellular automata to all disciplines of science (sociology, biology, physics, mathematics, economy)(collected papers in [4]).

In this paper, we discuss a social behavior model in cellular automata. This model came as a result of comprehensive research in dynamical structures of networks as these were constructed out of game theory and statistical physics. The rule is an outgrowth of the Prisoner's Dilemma game as it is stated by Nowak et al. A detailed numerical investigation of this model was first done in [3]. In that work, a different version of the rule is discussed, from a Game Theory perspective. In this work the model is redefined as a cellular automaton whose dynamics are discussed. Some preliminary results of this approach are presented in [1].

The model implements the following scenario: Static nodes on the grid interact among each other alternating between two states, according to a cost table that defines the e®ect of the interstate relationship among neighboring nodes. By the end of the round each node carries a reward and it's new state equals the state of the node that gets the highest reward
in the neighborhood. The model is implemented in one and two dimensional space and initiates from simple or disordered initial conditions. It's behavior is observed and classi¯ed as it's reward parameters vary. With the use of result from cellular automata and chaos theory we thoroughly analyze and discuss the amount of complexity this rule produces.
From the statistical physics point of view one can connect this model with the Ising Model where global state (i.e. the state of each node) is a random variable following a probability distribution that is characterized by the state condition among neighboring nodes.

It can be considered as a model of social as well as economic, biological and other complex networks.

Keywords : Complex Dynamics, Cellular Automata, Chaos, Self-Organization, Social Network Models

References
[1] G.P. Papavassilopoulos C.E Somarakis and F.E. Udwadia. A dynamic rule in cellular automata. In 22nd European Conference on Modelling and Simulation, pages 164{170, 2008.
[2] J. Von Neumann. The general and logical theory of automata. Celebral Mechanisms in Behavior - The Hixon Symposium, pages 1{41, 1948.
[3] R.J. Wiederien and F.E. Udwadia. Global patterns from local interactions. Int. J. Bifurcation and Chaos, 14(8):2555{2578, 2004.
[4] S. Wolfram. Automata and Complexity. Collected Papers. Addison-Wesley, 2000.