The Cavity Approach to Parallel Dynamics of Ising Spins on a Graph

TL;DR

This paper applies the cavity method to analyze the parallel dynamics of disordered Ising models on graphs, deriving recursive equations for single-site probabilities and exploring phase diagrams for various graph structures.

Contribution

It introduces a set of cavity equations for non-equilibrium dynamics of Ising models on graphs, extending existing methods to asymmetric and partially symmetric networks.

Findings

Exact solutions for directed graphs' stationary distributions

Phase diagrams for Ising models on asymmetric Bethe lattices

Validation of theoretical predictions through simulations

Abstract

We use the cavity method to study parallel dynamics of disordered Ising models on a graph. In particular, we derive a set of recursive equations in single site probabilities of paths propagating along the edges of the graph. These equations are analogous to the cavity equations for equilibrium models and are exact on a tree. On graphs with exclusively directed edges we find an exact expression for the stationary distribution of the spins. We present the phase diagrams for an Ising model on an asymmetric Bethe lattice and for a neural network with Hebbian interactions on an asymmetric scale-free graph. For graphs with a nonzero fraction of symmetric edges the equations can be solved for a finite number of time steps. Theoretical predictions are confirmed by simulation results. Using a heuristic method, the cavity equations are extended to a set of equations that determine the marginals of the stationary distribution of Ising models on graphs with a nonzero fraction of symmetric edges. The results of this method are discussed and compared with simulations.