Three lemmas on the dynamic cavity method

TL;DR

This paper investigates the dynamic cavity method for dilute kinetic Ising models, revealing conditions under which the equations simplify or require additional assumptions, with implications for understanding model dynamics.

Contribution

It presents three lemmas clarifying the behavior of the dynamic cavity method under different update rules and symmetry conditions in dilute kinetic Ising models.

Findings

For fully asymmetric models, the equations reduce to a Markovian dynamics.

Sequential updates do not lead to Markovian dynamics without time factorization.

Fixed points of Belief propagation coincide with dynamic cavity fixed points in symmetric models.

Abstract

We study the dynamic cavity method for dilute kinetic Ising models with synchronous update rules. For the parallel update rule we find for fully asymmetric models that the dynamic cavity equations reduce to a Markovian dynamics of the (time-dependent) marginal probabilities. For the random sequential update rule, also an instantiation of a synchronous update rule, we find on the other hand that the dynamic cavity equations do not reduce to a Markovian dynamics, unless an additional assumption of time factorization is introduced. For symmetric models we show that a fixed point of ordinary Belief propagation is also a fixed point of the dynamic cavity equations in the time factorized approximation. For clarity, the conclusions of the paper are formulated as three lemmas.