Rate Equations for Graphs

TL;DR

This paper introduces a framework combining graph transformation systems and mean field approximations to automatically generate rate equations for stochastic graph models, enabling advanced analysis of complex dynamics.

Contribution

It develops a novel, automated method to derive rate equations and mean field approximations for stochastic graph transformation systems, expanding analytical capabilities.

Findings

Derived an expression for the mean velocity of a two-legged walker protein on DNA.

Automated the derivation of rate equations and approximations for complex graph models.

Provided an implementation and example models accessible online.

Abstract

In this paper, we combine ideas from two different scientific traditions: 1) graph transformation systems (GTSs) stemming from the theory of formal languages and concurrency, and 2) mean field approximations (MFAs), a collection of approximation techniques ubiquitous in the study of complex dynamics. Using existing tools from algebraic graph rewriting, as well as new ones, we build a framework which generates rate equations for stochastic GTSs and from which one can derive MFAs of any order (no longer limited to the humanly computable). The procedure for deriving rate equations and their approximations can be automated. An implementation and example models are available online at https://rhz.github.io/fragger. We apply our techniques and tools to derive an expression for the mean velocity of a two-legged walker protein on DNA.