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

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.

Keywords

Interaction networks; chemical reactions; injectivity; SR graph; network structure; multiple equilibria

2010 Mathematics Subject Classification

05C38, 05C50, 15A15, 34C99

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