Markovian Dynamics on Complex Reaction Networks

TL;DR

This paper reviews recent methods for analyzing Markovian reaction networks across various scientific fields, focusing on computational techniques, stationary behavior, and thermodynamic perspectives, with applications in opinion formation, gene regulation, and neural dynamics.

Contribution

It provides a comprehensive overview of modeling, computational, and theoretical approaches for Markovian reaction networks, highlighting recent advances and applications.

Findings

Development of numerical techniques for solving master equations

Introduction of potential energy landscape approach for stationary analysis

Exploration of thermodynamic analysis in reaction networks

Abstract

Complex networks, comprised of individual elements that interact with each other through reaction channels, are ubiquitous across many scientific and engineering disciplines. Examples include biochemical, pharmacokinetic, epidemiological, ecological, social, neural, and multi-agent networks. A common approach to modeling such networks is by a master equation that governs the dynamic evolution of the joint probability mass function of the underling population process and naturally leads to Markovian dynamics for such process. Due however to the nonlinear nature of most reactions, the computation and analysis of the resulting stochastic population dynamics is a difficult task. This review article provides a coherent and comprehensive coverage of recently developed approaches and methods to tackle this problem. After reviewing a general framework for modeling Markovian reaction networks and giving specific examples, the authors present numerical and computational techniques capable of evaluating or approximating the solution of the master equation, discuss a recently developed approach for studying the stationary behavior of Markovian reaction networks using a potential energy landscape perspective, and provide an introduction to the emerging theory of thermodynamic analysis of such networks. Three representative problems of opinion formation, transcription regulation, and neural network dynamics are used as illustrative examples.