From Combinatorics to Geometry: The Dynamics of Graph Gradient Diffusion

Abstract: We discuss a link between graph theory and geometry that arises when considering graph dynamical systems with odd interactions. The equilibrium set in such systems is not a collection of isolated points, but rather a union of manifolds, which may intersect creating singularities and may vary in dimension. We prove that geometry and stability of such manifolds are governed by combinatorial properties of the underlying graph. In particular, we derive an upper bound on the dimension of the equilibrium set using graph homology and a lower bound using graph coverings. Moreover, we show how graph automorphisms relate to geometric singularities and prove that the decomposition of a graph into $2$-vertex-connected components induces a decomposition of the equilibrium set that preserves three notions of stability.