Dynamic message-passing equations for models with unidirectional dynamics

TL;DR

This paper develops a general method using dynamic cavity equations to analyze unidirectional models like epidemic and rumor spreading on complex networks, providing accurate approximations for their dynamics.

Contribution

The authors introduce a novel dynamic message-passing framework for unidirectional models, applicable to arbitrary initial conditions and locally tree-like graphs, enhancing analysis of out-of-equilibrium systems.

Findings

Equations are asymptotically exact on tree-like graphs.

Method provides good approximations on real-world networks.

Applicable to models like Ising, SIR, and rumor spreading.

Abstract

Understanding and quantifying the dynamics of disordered out-of-equilibrium models is an important problem in many branches of science. Using the dynamic cavity method on time trajectories, we construct a general procedure for deriving the dynamic message-passing equations for a large class of models with unidirectional dynamics, which includes the zero-temperature random field Ising model, the susceptible-infected-recovered model, and rumor spreading models. We show that unidirectionality of the dynamics is the key ingredient that makes the problem solvable. These equations are applicable to single instances of the corresponding problems with arbitrary initial conditions, and are asymptotically exact for problems defined on locally tree-like graphs. When applied to real-world networks, they generically provide a good analytic approximation of the real dynamics.