A matrix product algorithm for stochastic dynamics on networks, applied to non-equilibrium Glauber dynamics

TL;DR

This paper presents a novel matrix product algorithm for simulating stochastic dynamics on networks, offering improved accuracy and efficiency over Monte Carlo methods, especially for non-equilibrium Glauber dynamics in the kinetic Ising model.

Contribution

The paper introduces a matrix product approximation method for edge messages in stochastic network dynamics, extending cavity methods to handle cycles and improve simulation accuracy.

Findings

Better error scaling than Monte Carlo methods

Accurate evaluation of small expectation value observables

Effective for both single instances and the thermodynamic limit

Abstract

We introduce and apply a novel efficient method for the precise simulation of stochastic dynamical processes on locally tree-like graphs. Networks with cycles are treated in the framework of the cavity method. Such models correspond, for example, to spin-glass systems, Boolean networks, neural networks, or other technological, biological, and social networks. Building upon ideas from quantum many-body theory, the new approach is based on a matrix product approximation of the so-called edge messages -- conditional probabilities of vertex variable trajectories. Computation costs and accuracy can be tuned by controlling the matrix dimensions of the matrix product edge messages (MPEM) in truncations. In contrast to Monte Carlo simulations, the algorithm has a better error scaling and works for both, single instances as well as the thermodynamic limit. We employ it to examine prototypical non-equilibrium Glauber dynamics in the kinetic Ising model. Because of the absence of cancellation effects, observables with small expectation values can be evaluated accurately, allowing for the study of decay processes and temporal correlations.