Parallel dynamics of disordered Ising spin systems on finitely connected directed random graphs with arbitrary degree distributions

TL;DR

This paper analyzes the stochastic parallel dynamics of disordered Ising spin systems on directed random graphs with arbitrary degree distributions, providing exact solutions for asymmetric graphs and approximations for more general cases.

Contribution

It offers a comprehensive analytical framework for understanding Ising spin dynamics on complex directed graphs, including exact solutions and approximations for various graph symmetries.

Findings

Exact dynamics for fully asymmetric graphs.

Approximate stationary solutions for symmetric graphs.

Analytical methods applicable to complex network structures.

Abstract

We study the stochastic parallel dynamics of Ising spin systems defined on finitely connected directed random graphs with arbitrary degree distributions, using generating functional analysis. For fully asymmetric graphs the dynamics of the system can be completely solved, due to the asymptotic absence of loops. For arbitrary graph symmetry, we solve the dynamics exactly for the first few time steps, and we construct approximate stationary solutions.