Order parameter allows classification of planar graphs based on balanced fixed points in the Kuramoto model

Phase balanced states are a highly underexplored class of solutions of the Kuramoto model and other coupled oscillator models on networks. So far, coupled oscillator research focused on phase synchronized solutions. Yet, global constraints on oscillators may forbid synchronized state, rendering phase balanced states as the relevant stable state. If, for example, oscillators are driving the contractions of a fluid filled volume, conservation of fluid volume constrains oscillators to balanced states as characterized by a vanishing Kuramoto order parameter. It has previously been shown that stable, balanced patterns in the Kuramoto model exist on circulant graphs. However, which noncirculant graphs first of all allow for balanced states and what characterizes the balanced states is unknown. Here, we derive rules of how to build noncirculant, planar graphs allowing for balanced states from the simple cycle graph by adding loops or edges to it. We thereby identify different classes of small planar networks allowing for balanced states. Investigating the balanced states' characteristics, we find that the variance in basin stability scales linearly with the size of the graph for these networks. We introduce the balancing ratio as an order parameter based on the basin stability approach to classify balanced states on networks and evaluate it analytically for a subset of the network classes. Our results offer an analytical description of noncirculant graphs supporting stable, balanced states and may thereby help to understand the topological requirements on oscillator networks under global constraints.