Benign nonconvexity of synchronization landscape induced by graph skeletons

TL;DR

This paper investigates the optimization landscape of the Kuramoto model on graphs, revealing that certain structured graphs like quasi-threshold graphs exhibit benign landscapes where all second-order stationary points are globally optimal, facilitating synchronization.

Contribution

It uncovers a new mechanism for global synchronization on structured graphs via local synchronization propagating along skeletons, contrasting with previous approaches based on proximity to complete graphs.

Findings

Quasi-threshold graphs induce benign energy landscapes for synchronization.

Second-order stationary points correspond to global minima in these graphs.

Synchronization propagates sequentially along graph skeletons.

Abstract

We study the homogeneous Kuramoto model on a graph and the geometry of its underlying optimization landscape $\min_{\boldsymbol \theta \in \mathbb R^n}-\sum_{1\leq i,j\leq n} A_{ij}\cos(\theta_i-\theta_j).$ This problem admits a dual interpretation. On the one hand, it can be viewed as an unconstrained optimization problem, seeking configurations of points on the unit circle that minimize the energy function. On the other hand, the same function serves as a Lyapunov potential governing the dynamics of the homogeneous Kuramoto model. A central question is to identify which graphs induce a benign energy landscape, in the sense that every second-order stationary point is a global minimizer, corresponding to the fully synchronized state. In this case, the graph is said to be globally synchronizing. Most existing results establish global synchronization by exploiting the fact that the complete graph is globally synchronizing, and by showing that graphs sufficiently close to it inherit this property. In contrast, we uncover a fundamentally different mechanism: on highly-structured graph classes, namely quasi-threshold graphs, global synchronization unfolds through a sequential process of local synchronization that propagates along their underlying skeletons. Our approach relies on a detailed analysis of the phasor geometry at second-order stationary points of the nonconvex energy landscape.