Abstract
Conway’s Game of Life, which is one of the most studied cellular automata, contains a self-organized rest state with residual clusters of live cells including the well-known patterns like Block and Beehive. The present article aims to discuss the self-organizing process from the viewpoint of a network representation of the Game of Life. Scale-free property of the network associated with the rest state is resilient against consecutive removals of a hub node. Of the two types of residual clusters, whose links continue to grow or not over time, the type of clusters accompanied by growing links is essential in the self-organizing process. Mixing the two types of clusters, which diversifies scales of branch graphs that correspond to avalanches caused by one-cell perturbations in the rest state, contributes to the scale-free property of the rest state. Furthermore, a network of clusters can be obtained from the rest-state network by regarding the clusters as composites of live cells.
Highlights
In this article we studied the self-organizing process of Life through the network representation (NR) approach, which, unlike dot patterns, can reveal dynamical aspects because it contains a history of changes of cell states
Associated networks of clusters in a rest state can be classified into growth type and non-growth type according to the growth of their network
By analyzing the scale-free property of the rest-state network, we noticed that type-G clusters are essential in the self-organizing process to connect clusters through long-range links
Summary
Conway’s Game of Life, or Life, is one of the most famous cellular automata (CA) and is defined on a two-dimensional square lattice of cells, where each cell has two state values, true or false, associated with live or dead cells (Gardner, 1970; Berlekamp et al, 1982). The rest-state network, which encompasses the patterns’ networks and connects the patterns together, is a visualization of underlying tension that causes the growth of avalanches catalyzed by one-cell perturbations. A survey of the self-organizing process is given in the fourth section where a network connecting clusters and its degree distribution are discussed. Configuration of Life reaches a rest state that contains static and oscillating patterns (clusters) Moving clusters, such as Glider, may exist if they move along closed paths when periodic boundary conditions are adopted on the lattice. By using out-degree as an order parameter to verify SOC together with avalanche lifetime and size, Kayama (2013) investigated power-law scaling in rest-state networks.
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