Synchronization in random networks of identical phase oscillators: A graphon approach

TL;DR

The paper develops a graphon-based framework for random networks of identical phase oscillators and proves that solutions of the finite $W$-random network converge in $L^{\infty}$ to the continuum graphon system with high probability as $n\to\infty$ under regularity conditions on the graphon and in sparse regimes. This convergence allows transferring synchronization insights from the continuum model to large finite Erdős–Rényi networks, with explicit results for homogeneous Kuramoto and Sakaguchi–Kuramoto oscillators. The main contributions are the $L^{\infty}$-convergence theorem and its application to phase and frequency synchronization on random graphs, including explicit conditions for the Sakaguchi–Kuramoto case. The framework broadens the scope of synchronization analysis beyond gradient systems and provides a principled approach for studying large-scale random networks in physics, biology, and engineering.

Abstract

Networks of coupled nonlinear oscillators have been used to model circadian rhythms, flashing fireflies, Josephson junction arrays, high-voltage electric grids, and many other kinds of self-organizing systems. Recently, several authors have sought to understand how coupled oscillators behave when they interact according to a random graph. Here we consider interaction networks generated by a graphon model known as a $W$-random network, and examine the dynamics of an infinite number of identical phase oscillators. We show that with sufficient regularity on $W$, the solution to the dynamical system over a $W$-random network of size $n$ converges in the $L^{\infty}$ norm to the solution of the infinite graphon system, with high probability as $n\rightarrow\infty$. We leverage this convergence result to explore synchronization for two classes of identical phase oscillators on Erdős-Rényi random graphs. This result suggests a framework for studying synchronization properties in large but finite random networks.

Figures (2)

• Figure 1: Schematic illustration of how to obtain a random network from a graphon. In the example shown, a random graph on $5$ nodes is generated from the continuous graphon $W(x,y) = \sin(\pi{x})\sin(\pi{y})$. The discretized graphon is a function defined on the unit square, $I^{2}$, where each $x,y\in {I_i^{(5)}\times{I_j^{(5)}}}$ assumes the value $W^{5}_{ij} = 5^{2}\int_{I_i^{(5)}\times{I_j^{(5)}}}W(x,y)dydx$. From this discretized graphon, one may construct a random network of size $5$ by letting the probability that there exists an edge between nodes $i$ and $j$ be equal to $W^{(5)}_{ij}$.
• Figure 2: Snapshot of $n=20$ phase oscillators plotted on the unit circle after frequency synchronization has been achieved for Sakaguchi-Kuramoto oscillators (with $\beta=\frac{\pi}{50}$), interacting over an Erdős-Rényi random network for three values of the edge probability: $p=0.5, p=0.7$, and $p=0.9$. In all three instances, the order parameter (a scalar $r\in[0,1]$ indicating how aligned the phases are Dorfler2014) is close to $1$ and the oscillators are in nearly perfect phase alignment, suggesting a phenomenon beyond mere frequency synchronization.

Theorems & Definitions (34)

• Theorem 2.1
• proof
• Theorem 2.2
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• proof : Proof of Theorem \ref{['main_theorem']}
• Theorem 4.1
• proof