A new decomposition of the graph Laplacian and the binomial structure of mass-action systems

TL;DR

This paper introduces a novel graph Laplacian decomposition that enhances understanding of mass-action systems' stability, revealing new algebraic and geometric insights into their behavior.

Contribution

It presents a new algebraic decomposition of the graph Laplacian and applies it to analyze the stability and structure of mass-action systems in chemical networks.

Findings

Decomposition involves an invertible core matrix, tree constants, and an auxiliary graph incidence matrix.

Clarifies the binomial structure of weakly reversible mass-action systems.

Provides a polyhedral-geometry proof of asymptotic stability for complex-balanced systems.

Abstract

We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible $\textit{core matrix}$, the vector of tree constants, and the incidence matrix of an auxiliary graph, representing an order on the vertices. Depending on the particular order, the core matrix has additional properties. Our results are graph-theoretic/algebraic in nature. As a first application, we further clarify the binomial structure of (weakly reversible) mass-action systems, arising from chemical reaction networks. Second, we extend a classical result by Horn and Jackson on the asymptotic stability of special steady states (complex-balanced equilibria). Here, the new decomposition of the graph Laplacian allows us to consider regions in the positive orthant with given $\textit{monomial evaluation orders}$ (and corresponding polyhedral cones in logarithmic coordinates). As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain $\textit{binomial differential inclusions}$. In particular, this holds for complex-balanced mass-action systems, and hence we also obtain a polyhedral-geometry proof of the classical result.