Paper
Threshold Graphs Are Globally Synchronizing
Authors
Hongjin Wu, Ulrik Brandes
Abstract
The Kuramoto model describes phase oscillators on the unit circle whose interactions are encoded by a graph. Each edge acts like a spring that pulls the two adjacent oscillators toward each other whenever their phases differ. A central question is to determine which graphs are globally synchronizing, meaning that trajectories of the Kuramoto dynamics converge to the fully synchronized state from almost all initial conditions, except for a set of measure zero. This property is tightly linked to the benign nonconvexity of the model's energy landscape. Existing guarantees for global synchronization rely on minimum-degree thresholds, which require the graph to be highly dense. In this work, we show that connected threshold graphs, whose density may vary from to , are globally synchronizing. Our proof relies on a phasor-geometric analysis of the stationary points of the associated energy landscape.