The realization of admissible graphs for coupled vector fields

In a coupled network cells can interact in several ways. There is a vast literature from the last twenty years that investigates this interacting dynamics under a graph theory formalism, namely as a graph endowed with an input-equivalence relation on the set of vertices that enables a characterization of the admissible vector fields that rules the network dynamics. The present work goes in the direction of answering an inverse problem: for $n \geq 2$, any mapping on $\mathbb{R}^n$ can be realized as an admissible vector field for some graph with the number of vertices depending on (but not necessarily equal to) $n$. Given a mapping, we present a procedure to construct all non-equivalent admissible graphs, up to the appropriate equivalence relation. We also give an upper bound for the number of such graphs. As a consequence, invariant subspaces under the vector field can be investigated as the locus of synchrony states supported by an admissible graph, in the sense that a suitable graph can be chosen to realize couplings with more (or less) synchrony than another graph admissible to the same vector field. The approach provides in particular a systematic investigation of occurrence of chimera states in a network of van der Pol identical oscillators.