Structurally Dynamic Cellular Automata Andrew Ilachinski Center for Naval Analyses, Alexandria, VA, USA Article Outline Glossary Definition of the Subject Introduction The Basic Model Emerging Patterns and Behaviors SDCA as Models of Computation Generalized SDCA Models Related Graph Dynamical Systems SDCA as Models of Fundamental Physics Future Directions and Speculations Bibliography Glossary Adjacency matrix The adjacency matrix of a graph with N sites is an N  N matrix [a ij ] with entries a ij = 1 if i and j are linked, and a ij = 0 otherwise. The adjacency matrix is symmetric (a ij = a ji ) if the links in the graph are undirected. Coupler link rules Coupler rules are local rules that act on pairs of next-nearest sites of a graph at time t to decide whether they should be linked at t + 1. The decision rules fall into one of three basic classes – totalistic (T), outer-totalistic (OT) or restricted-totalistic (RT) – but can be as varied as those for con- ventional cellular automata. Decoupler link rules Decoupler rules are local rules that act on pairs of linked sites of a graph at time t to decide whether they should be unlinked at t + 1. As for coupler rules, the decision rules fall into one of three basic classes – totalistic (T), outer-totalistic (OT) or restricted-totalistic (RT) – but can be as varied as those for conventional cellular automata. Degree The degree of a node (or site, i) of a graph is equal to the number of distinct nodes to which i is linked, and where the links are assumed to possess no directional information. In general graphs, the in-degree (= number of incoming links towards i) is distinguished from the out-degree (= number of outgoing links originating at i). Effective dimension A quantity used to approx- imate the dimensionality of a graph. It is defined as the ratio between the average num- ber of next-nearest neighbors to the average degree, both averaged over all nodes of the graph. The effective dimension equals the Euclidean dimension d, in cases where the graph is the familiar d-dimensional hypercubic lattice. Graph A graph is a finite, nonempty set of nodes (referred to as “sites” throughout this article), together with (a possibly empty) set of edges (or links). The links may be either directed (in which case the edge from a site i, say, is directed away from i toward another site j, and is considered distinct from another directed edge originating at j and pointed toward i) or undirected (in which case if a link exists between sites i and j it carries no directional information). Graph grammar Graph grammars (sometimes also referred to as graph rewriting systems) apply formal language theory to networks. Each language specifies the space of “valid structures”, and the production (or “rewrite”) rules by which given graphs may be trans- formed into other valid graphs. Graph metric function The graph metric func- tion defines the distance between any two nodes, i and j. It is equal to the length of the shortest path between i and j. If no path exists (such as when i and j are on two disconnected components of the same graph), the distance is assumed to be equal to 1. # Springer-Verlag 2009 A. Adamatzky (ed.), Cellular Automata, https://doi.org/10.1007/978-1-4939-8700-9_528 Originally published in R. A. Meyers (ed.), Encyclopedia of Complexity and Systems Science, # Springer-Verlag 2009 https://doi.org/10.1007/978-0-387-30440-3_528 29