Network representation of the game of life and self-organized criticality

Abstract

Conway's Game of Life is one of the most famous and frequently studied cellular automata. This paper introduces a network representation of the Game of Life and studies its relation to self-organized criticality. Self-organized criticality in the Game of Life is reconfirmed by studying power-law scaling for the distributions of avalanche scales: lifetimes, sizes, and out-degrees in a rest-state network. Avalanches are caused by one-cell perturbations of the rest state. Finite-size scaling analysis shows that avalanche lifetime and out-degree can be regarded as order parameters with characteristic lengths dependent on lattice size. The rest-state network of the Game of Life expresses a power-law degree distribution of out-links with a cut-off. Rule T52 of the one-dimensional binary five-neighbor totalistic cellular automata is also discussed in terms of the out-degrees of the rest-state network. Network representations of binary cellular automata can be used to assess their self-organized criticality.