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Synchronization in systems of coupled Kuramoto oscillators may depend ontheir natural frequencies, coupling, and underlying networks. In this paper, wereduce the alternatives to only one by considering identical oscillators wherethe only parameter that is allowed to change is the underlying network. Whilesuch a model was analyzed over the past few decades by studying the size of thebasin of attraction of the synchronized state on restricted families of graphs,here we address a qualitative question on general graphs. In an analogy toresistive networks with current sources, we describe an algorithm that producesinitial conditions that are often outside of the basin of attraction of thesynchronized state. In particular, if a graph allows a cyclic graph clusteringwith a sufficient number of clusters or contains a sufficiently long inducedsubpath without cut vertices of the graph then there is a non-synchronousstable phase-locked solution. Thus, we provide a partial answer to when thesynchronized state is not globally stable.

When is sync globally stable in sparse networks of identical Kuramoto oscillators?