Graph-theoretic approaches to injectivity and multiple equilibria in systems of interacting elements

TL;DR

This paper generalizes injectivity analysis from chemical reaction networks to broader interaction networks using graph-theoretic conditions, notably employing the DSR graph, to determine when systems have unique equilibria.

Contribution

It introduces a new graph-theoretic framework, including the DSR graph, to analyze injectivity in general interaction networks, extending previous chemical network results.

Findings

Graph-theoretic conditions can guarantee injectivity in a wide class of systems.

The DSR graph effectively represents interaction networks for injectivity analysis.

The proposed conditions are simple to state and verify.

Abstract

We extend previous work on injectivity in chemical reaction networks to general interaction networks. Matrix- and graph-theoretic conditions for injectivity of these systems are presented. A particular signed, directed, labelled, bipartite multigraph, termed the ``DSR graph'', is shown to be a useful representation of an interaction network when discussing questions of injectivity. A graph-theoretic condition, developed previously in the context of chemical reaction networks, is shown to be sufficient to guarantee injectivity for a large class of systems. The graph-theoretic condition is simple to state and often easy to check. Examples are presented to illustrate the wide applicability of the theory developed.