Global synchronization beyond dense graphs: the case of threshold graphs

Abstract

Given a graph \(G\) with adjacency matrix \(A\), consider the homogeneous Kuramoto energy $E_G(\boldsymbolθ):=\frac{1}{2}\sum_{1\leq i,j\leq n}A_{ij}\bigl(1-\cos(θ_i-θ_j)\bigr)$. We call \(G\) \emph{second-order globally synchronizing} if every second-order stationary point of \(E_G\) is fully synchronized. This property implies \emph{global synchronization}, namely that, up to a measure-zero set of initial conditions, trajectories of the Kuramoto model converge to a fully synchronized state. A fundamental graph-theoretic question is to identify which graph structures have this property. Existing guarantees for global synchronization typically require large minimum degree which forces the graph to be very dense, or good expansion properties. In this paper, we show that synchronization can also arise from a different, purely structural mechanism. More precisely, we prove that threshold graphs, a classical recursively defined graph class, are second-order globally synchronizing, and hence globally synchronizing. Thus, globally synchronizing graphs need not be very dense, have large minimum degree, or satisfy strong expansion-type conditions. The proof exploits the recursive construction of threshold graphs: local phasor constraints imposed by second-order stationarity are propagated along the construction sequence until full synchronization is forced.