From Network Reliability to the Ising Model: A Parallel Scheme for Estimating the Joint Density of States

TL;DR

This paper presents a novel parallel Monte Carlo scheme to efficiently estimate the joint density of states for the Ising model, enabling practical computation of network reliability related to Ising feasibility on complex networks.

Contribution

It introduces a new parallel Markov chain Monte Carlo method for estimating the joint density of states, improving the simulation of the Ising model's partition function on arbitrary networks.

Findings

Enables simulation of the Ising model with external fields on complex networks.

Provides an efficient approximation of the partition function for large state spaces.

Addresses limitations of naive Monte Carlo sampling in this context.

Abstract

Network reliability is the probability that a dynamical system composed of discrete elements interacting on a network will be found in a configuration that satisfies a particular property. We introduce a new reliability property, Ising feasibility, for which the network reliability is the Ising model s partition function. As shown by Moore and Shannon, the network reliability can be separated into two factors: structural, solely determined by the network topology, and dynamical, determined by the underlying dynamics. In this case, the structural factor is known as the joint density of states. Using methods developed to approximate the structural factor for other reliability properties, we simulate the joint density of states, yielding an approximation for the partition function. Based on a detailed examination of why naive Monte Carlo sampling gives a poor approximation, we introduce a novel parallel scheme for estimating the joint density of states using a Markov chain Monte Carlo method with a spin exchange random walk. This parallel scheme makes simulating the Ising model in the presence of an external field practical on small computer clusters for networks with arbitrary topology with 10 to 6 energy levels and more than 10 to 308 microstates.