A Graph-Theoretical Approach for the Analysis and Model Reduction of Complex-Balanced Chemical Reaction Networks

TL;DR

This paper introduces a graph-theoretical framework for analyzing complex-balanced chemical reaction networks, enabling characterization of equilibria, stability analysis, and a novel model reduction technique based on Laplacian matrices.

Contribution

It presents a new mathematical formulation using graph theory and Laplacian matrices for complex-balanced networks, including a model reduction method analogous to Kron reduction.

Findings

Characterizes all equilibria of complex-balanced networks

Provides stability analysis methods for these networks

Proposes a Schur complement-based model reduction technique

Abstract

In this paper we derive a compact mathematical formulation describing the dynamics of chemical reaction networks that are complex-balanced and are governed by mass action kinetics. The formulation is based on the graph of (substrate and product) complexes and the stoichiometric information of these complexes, and crucially uses a balanced weighted Laplacian matrix. It is shown that this formulation leads to elegant methods for characterizing the space of all equilibria for complex-balanced networks and for deriving stability properties of such networks. We propose a method for model reduction of complex-balanced networks, which is similar to the Kron reduction method for electrical networks and involves the computation of Schur complements of the balanced weighted Laplacian matrix.