On the Connectivity of the Disguised Toric Locus of a Reaction Network

TL;DR

This paper investigates the structure of the parameter space of reaction networks that can be realized as complex-balanced systems, proving that the set of such parameters is path-connected and exploring its topological properties.

Contribution

It establishes the path-connectedness of the disguised toric locus and its real extension, and shows the closure contains unions of subnetworks' loci, advancing understanding of reaction network parameter spaces.

Findings

Disguised toric locus is path-connected.

The real disguised toric locus is path-connected.

Closure contains union of subnetworks' loci.

Abstract

Complex-balanced mass-action systems are some of the most important types of mathematical models of reaction networks, due to their widespread use in applications, as well as their remarkable stability properties. We study the set of positive parameter values (i.e., reaction rate constants) of a reaction network $G$ that, according to mass-action kinetics, generate dynamical systems that can be realized as complex-balanced systems, possibly by using a different graph $G'$. This set of parameter values is called the disguised toric locus of $G$. The $\mathbb{R}$-disguised toric locus of $G$ is defined analogously, except that the parameter values are allowed to take on any real values. We prove that the disguised toric locus of $G$ is path-connected, and the $\mathbb{R}$-disguised toric locus of $G$ is also path-connected. We also show that the closure of the disguised toric locus of a reaction network contains the union of the disguised toric loci of all its subnetworks.