Dynamical replica analysis of processes on finitely connected random graphs I: vertex covering

TL;DR

This paper develops a dynamical replica analysis framework for spin models on finitely connected random graphs, applying it to vertex cover optimization and validating predictions with simulations.

Contribution

It introduces a general dynamical replica method for complex networks and applies it to vertex cover, achieving near-perfect agreement with simulations.

Findings

Theoretical predictions match Monte Carlo simulations closely.

The method provides detailed joint spin-field probability dynamics.

Application to vertex cover demonstrates the approach's effectiveness.

Abstract

We study the stochastic dynamics of Ising spin models with random bonds, interacting on finitely connected Poissonnian random graphs. We use the dynamical replica method to derive closed dynamical equations for the joint spin-field probability distribution, and solve these within the replica symmetry ansatz. Although the theory is developed in a general setting, with a view to future applications in various other fields, in this paper we apply it mainly to the dynamics of the Glauber algorithm (extended with cooling schedules) when running on the so-called vertex cover optimization problem. Our theoretical predictions are tested against both Monte Carlo simulations and known results from equilibrium studies. In contrast to previous dynamical analyses based on deriving closed equations for only a small numbers of scalar order parameters, the agreement between theory and experiment in the present study is nearly perfect.