Benign Nonconvexity of Synchronization Landscape Induced by Graph Skeletons

TL;DR

The paper investigates benign nonconvexity in the energy landscape of the homogeneous Kuramoto model on graphs, showing that certain graphs—specifically connected quasi-threshold graphs—exhibit global synchronization such that every second-order stationary point is fully synchronized. It develops a geometric framework based on phasor geometry and introducing geometric twins to analyze SOSP, and shows a local-to-global synchronization mechanism that propagates along the graph skeleton from leaves upward. The key contribution is proving global synchrony for this graph class and revealing a propagation principle that does not rely on proximity to the complete graph, thereby enriching the understanding of when simple local dynamics yield global optimality. The results have implications for designing and analyzing networked oscillator systems where synchronization emerges through structured, skeleton-based propagation rather than dense connectivity.

Abstract

We consider the homogeneous Kuramoto model on a graph and study the geometry of its associated nonconvex energy landscape. This problem admits a dual interpretation. On the one hand, it can be viewed as a geometric optimization problem, seeking configurations of phases that minimize the energy function $E(\boldsymbolθ):=-\sum_{1\leq i,j\neq n}A_{ij}\cos(θ_i-θ_j)$. On the other hand, the same function serves as the potential governing the dynamics of the classical 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 relating a given graph to the complete graph, which is known to be globally synchronizing, and by showing that graphs sufficiently close to it inherit this property. In contrast, we uncover a fundamentally different mechanism: global synchronization, despite being a collective phenomenon, unfolds on these graphs through a sequential process of local synchronization that propagates along their structural skeletons. Our approach is based on a detailed analysis of the phasor geometry at second-order stationary points of the nonconvex energy landscape.

Figures (12)

• Figure 1: Threshold graph defined by comparability graph (right) of rooted caterpillar (left).
• Figure 2: A quasi-threshold graph as the comparability graph of a rooted tree (with root $A$). The vertex set is ordered by the rooted tree structure, with edges corresponding to comparable pairs.
• Figure 3: Feasible region of strengths $(\mu_a,\mu_b)$ for geometric open twins. It lies on the lines $\mu_a = \mu_b$ (black, corresponding to $a$ and $b$ synchronized) and $\mu_a = -\mu_b$ (red, corresponding to $a$ and $b$ antipodal). At their intersection $(0,0)$, marked by a hollow dot, there is no additional constraint on the phasors of $a$ and $b$.
• Figure 4: Feasible region of strengths $(\mu_a,\mu_b)$ for geometric closed twins. It lies on the lines $\mu_a = \mu_b$ (black, with $a$ and $b$ synchronized) and $\mu_a + \mu_b = -2$ (red, with $a$ and $b$ in antipodal position). At their intersection $(-1,-1)$, marked by a hollow dot, there is no additional constraint on the phasors of $a$ and $b$.
• Figure 5: Structural closed twins attaining nodewise equilibrium.

Theorems & Definitions (19)

• Theorem 1.1
• Lemma 2.1
• Lemma 2.2
• Definition 4.1: Quasi-threshold graph
• Theorem 5.1: Geometric Open Twins Lemma
• proof
• Theorem 5.2: Geometric Stable Closed Twins
• proof
• Remark 6.1
• Corollary 6.2