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There are a number of models of coupled oscillator networks where thequestion of the stability of fixed points reduces to calculating the index of agraph Laplacian. Some examples of such models include the Kuramoto andKuramoto--Sakaguchi equations as well as the swing equations, which govern thebehavior of generators coupled in an electrical network. We show that the indexcalculation can be related to a dual calculation which is done on the firsthomology group of the graph, rather than the vertex space. We also show thatthis representation is computationally attractive for relatively sparse graphs,where the dimension of the first homology group is low, as is true in manyapplications. We also give explicit formulae for the dimension of the unstablemanifold to a phase-locked solution for graphs containing one or two loops. Asan application, we present some novel results for the Kuramoto model defined ona ring and compute the longest possible edge length for a stable solution.

Graph Homology and Stability of Coupled Oscillator Networks