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GCANs: Global Cellular Automaton Networks

Seth J. Chandler
Organization: Unversity of Houston Law Center

2004 International Mathematica Symposium
Conference location

Banff, Canada

Many social, economic, biological and legal interactions can be thought of as situations in which actors have private information, public signals that they can give off, and various algorithms for figuring out which signals to emit. Often those algorithms involve introspection a look at one's own condition and values and an examination of the behavior of "neighboring" members of society. This article conceptualizes something known as a Global Cellular Automaton Network (GCAN) as a general tool for exploring systems that feature distributed information (with varying degrees of privacy), potentially heterogeneous values, and complex patterns of connection between agents. It does so using the Mathematica programming language and drawing on some of the insights of Stephen Wolfram's book, A New Kind of Science [Wolfram 2002]. The GCAN derives from the somewhat simpler construct of a global cellular automaton (GCA). A GCA is a single cellular automaton in which the evolution of each site is a function not only of the values of the site's neighbors which is the distinguishing feature of conventional cellular automata but also of certain global features of the automaton. A GCAN preserves the notion of evolution that depends in some way on global features of the system but further extends from both a cellular automaton and a GCA in at least two ways: (1) there are multiple cellular automata that are networked with each other; (2) each site within each cellular automaton node within the GCAN evolves, based not only on the values of its neighbors and on its own global characteristics but also on certain characteristics of the cellular automata to which it is connected. This extension creates a tool capable of great flexibility in the systematic study of complex systems. Moreover, GCANs can be used to explore "Random Cellular Networks", in which the update rule to be applied by each site on each iteration is determined by application of a node-specific cellular automaton to the signals being emitted by its neighboring nodes. They provide a vehicle for studying composition of cellular automata. This notebook begins the study of GCANs and shows how they can be implemented in Mathematica. The code for this versatile mechanism appears to be relatively speedy and almost as general as Mathematica's underlying CellularAutomaton function.

*Mathematics > Discrete Mathematics > Cellular Automata

cellular automata, A New Kind of Science, legal interactions, social interactions, economic interactions, GCAN, global cellular automaton network, distributed information
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