Mean field dynamics of graphs I: Evolution of probabilistic cellular automata for random and small-world graphs

TL;DR

This paper develops a mean field approach to analyze the dynamics of probabilistic cellular automata on random and small-world graphs, providing insights into phase transitions and complex behaviors in network models relevant to psychopathology.

Contribution

It extends mean field theory to non-regular graphs, enabling analysis of complex network dynamics and phase transitions in probabilistic cellular automata.

Findings

Mean field approximation accurately predicts dynamics across different graph sizes.

Bifurcation diagrams reveal phase transitions in network behavior.

The approach offers explanations for sudden behavioral changes in depression models.

Abstract

It was recently shown how graphs can be used to provide descriptions of psychopathologies, where symptoms of, say, depression, affect each other and certain configurations determine whether someone could fall into a sudden depression. To analyse changes over time and characterise possible future behaviour is rather difficult for large graphs. We describe the dynamics of networks using one-dimensional discrete time dynamical systems theory obtained from a mean field approach to (elementary) probabilistic cellular automata (PCA). Often the mean field approach is used on a regular graph (a grid or torus) where each node has the same number of edges and the same probability of becoming active. We show that we can use variations of the mean field of the grid to describe the dynamics of the PCA on a random and small-world graph. Bifurcation diagrams for the mean field of the grid, random, and small-world graphs indicate possible phase transitions for certain parameter settings. Extensive simulations indicate for different graph sizes (number of nodes) that the mean field approximation is accurate. The mean field approach allows us to provide possible explanations of 'jumping' behaviour in depression.