Mean field dynamics of stochastic cellular automata for random and small-world graphs

We aim to provide a theoretical framework to explain the discrete transitions of mood connecting ideas from network theory and dynamical systems theory. It was recently shown how networks (graphs) can be used to represent psychopathologies, where symptoms of, say, depression, affect each other and certain configurations determine whether someone could transition into a depression. To analyse changes over time and characterise possible future behaviour is in general rather difficult for large graphs. We describe the dynamics of graphs using one-dimensional discrete time dynamical systems theory obtained from a mean field approximation to stochastic cellular automata (SCA). Often the mean field approximation 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 provide quantitative results on the accuracy of using the mean field approximation for the grid and random and small-world graph to describe the dynamics of the SCA. Bifurcation diagrams for the mean field of the different graphs indicate possible phase transitions for certain parameter settings of the mean field. Simulations confirm for different graph sizes (number of nodes) that the mean field approximation is accurate.