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We discuss a link between graph theory and geometry that arises whenconsidering graph dynamical systems with odd interactions. The equilibrium setin such systems is not a collection of isolated points, but rather a union ofmanifolds, which may intersect creating singularities and may vary indimension. We prove that geometry and stability of such manifolds are governedby combinatorial properties of the underlying graph. In particular, we derivean upper bound on the dimension of the equilibrium set using graph homology anda lower bound using graph coverings. Moreover, we show how graph automorphismsrelate to geometric singularities and prove that the decomposition of a graphinto $2$-vertex-connected components induces a decomposition of the equilibriumset that preserves three notions of stability.

From Combinatorics to Geometry: The Dynamics of Graph Gradient Diffusion