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Phase balanced states are a highly under-explored class of solutions of theKuramoto model and other coupled oscillator models on networks. So far, coupledoscillator research focused on phase synchronized solutions. Yet, globalconstraints on oscillators may forbid synchronized state, rendering phasebalanced states as the relevant stable state. If for example oscillators aredriving the contractions of a fluid filled volume, conservation of fluid volumeconstraints oscillators to balanced states as characterized by a vanishingKuramoto order parameter. It has previously been shown that stable, balancedpatterns in the Kuramoto model exist on circulant graphs. However, whichnon-circulant graphs first of all allow for balanced states and whatcharacterizes the balanced states is unknown. Here, we derive rules of how tobuild non-circulant, planar graphs allowing for balanced states from the simplecycle graph by adding loops or edges to it. We thereby identify differentclasses of small planar networks allowing for balanced states. Investigatingthe balanced states' characteristics, we find that the variance in basinstability scales linearly with the size of the graph for these networks. Weintroduce the balancing ratio as a new order parameter based on the basinstability approach to classify balanced states on networks and evaluate itanalytically for a subset of the network classes. Our results offer ananalytical description of non-circulant graphs supporting stable, balancedstates and may thereby help to understand the topological requirements onoscillator networks under global constraints.

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