# The matrix product approximation for the dynamic cavity method

## TL;DR

This paper introduces a matrix product edge message algorithm for the dynamic cavity method, enabling efficient and accurate analysis of stochastic dynamics on tree-like graphs with controllable precision.

## Contribution

It presents a novel matrix product approximation technique for the dynamic cavity equations, improving computational efficiency and extending applicability to models with continuum time limits.

## Key findings

- Computation costs grow linearly with time for Glauber-Ising dynamics.
- The method outperforms Monte Carlo in error scaling, especially for low probability events.
- Optimized truncation schemes enhance accuracy and efficiency.

## Abstract

Stochastic dynamics of classical degrees of freedom, defined on vertices of locally tree-like graphs, can be studied in the framework of the dynamic cavity method which is exact for tree graphs. Such models correspond for example to spin-glass systems, Boolean networks, neural networks, and other technical, biological, and social networks. The central objects in the cavity method are edge messages -- conditional probabilities of two vertex variable trajectories. In this paper, we discuss a rather pedagogical derivation for the dynamic cavity method, give a detailed account of the novel matrix product edge message (MPEM) algorithm for the solution of the dynamic cavity equation as introduced in Phys. Rev. E 97, 010104(R) (2018), and present optimizations and extensions. Matrix product approximations of the edge messages are constructed recursively in an iteration over time. Computation costs and precision can be tuned by controlling the matrix dimensions of the MPEM in truncations. Without truncations, the dynamics is exact. Data for Glauber-Ising dynamics shows a linear growth of computation costs in time. In contrast to Monte Carlo simulations, the approach has a much better error scaling. Hence, it gives for example access to low probability events and decaying observables like temporal correlations. We discuss optimized truncation schemes and an extension that allows to capture models which have a continuum time limit.