Expander graphs are globally synchronizing

TL;DR

This work shows that strong graph expansion suffices to guarantee global synchronization in the homogeneous Kuramoto model. It develops an amplification framework tied to kernel stability and spectral graph theory to rule out spurious stable states on expanding graphs, and then applies it to Erdős–Rényi graphs near the connectivity threshold as well as Ramanujan and random regular graphs. The authors establish quantitative expansion-based criteria that imply global synchrony and prove that G(n,p) with $p \ge (1+\varepsilon)\frac{\log n}{n}$ is globally synchronizing with high probability, solving a conjecture about near-threshold graphs. Technical contributions include a novel kernel stability condition, a refined two-sided expansion analysis, and explicit random-matrix bounds for adjacency and Laplacian concentration, connecting energy landscapes to spectral properties of the network. These results demonstrate that sparse, pseudorandom graphs can robustly enforce global synchronization in network dynamics and related optimization landscapes.

Abstract

The Kuramoto model is fundamental to the study of synchronization. It consists of a collection of oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that a homogeneous Kuramoto model of identical oscillators on such a graph will converge to the fully synchronized state with all the oscillators having the same phase, for every initial state up to a set of measure zero. In particular, we show that for any $\varepsilon > 0$ and $p \geq (1 + \varepsilon) (\log n) / n$, the homogeneous Kuramoto model on the Erdős-Rényi random graph $G(n, p)$ is globally synchronizing with probability tending to one as $n$ goes to infinity. This improves on a previous result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu, and Bandeira. We also show that the model is globally synchronizing on any $d$-regular Ramanujan graph, and on typical $d$-regular graphs, for large enough degree $d$.

Figures (3)

• Figure 1: The function $s \colon (-\pi, \pi] \to \mathbb{R}$.
• Figure 2: Angles $\beta_{-1} \geqslant \beta_0 \geqslant \beta_1 \geqslant \dotsb \geqslant \beta_M \geqslant \beta_{M+1}$ in the amplification argument.
• Figure 3: The functions $c^-(\gamma)$ and $c^+(\gamma)$.

Theorems & Definitions (68)

• Definition 1.1: Globally synchronizing graph
• Theorem 1.2
• Definition 1.3
• Theorem 1.4
• Remark 1.5
• Corollary 1.6
• proof
• Remark 1.7
• Remark 1.8
• Definition 1.9: Spectral expander graph