Capturing the critical coupling of large random Kuramoto networks with graphons

TL;DR

This work develops a graphon-based mean-field framework to study synchronization in large random Kuramoto networks with random frequencies. It identifies a master nonlocal graphon equation F(u,Ω,K)=0 that governs existence and stability of synchronous states in the infinite-network limit and informs persistence to finite networks. For Erdős–Rényi networks, the onset of synchrony can arise either through co-dimension one saddle-node bifurcations or via bifurcations from the essential spectrum, depending on the frequency distribution. Center-manifold reductions for both the graphon and step-graphon problems yield reduced scalar dynamics with explicit coefficients that determine the bifurcation type and guarantee convergence K_{crit,n}→K_{crit} as n→∞. Overall, the framework links infinite-network graphon limits with finite random networks, enabling precise predictions of synchronization thresholds across network sizes.

Abstract

Collective oscillations and patterns of synchrony have long fascinated researchers in the applied sciences, particularly due to their far-reaching importance in chemistry, physics, and biology. The Kuramoto model has emerged as a prototypical mathematical equation to understand synchronization in coupled oscillators, allowing one to study the effect of different frequency distributions and connection networks between oscillators. In this work we provide a framework for determining both the emergence and the persistence of synchronous solutions to Kuramoto models on large random networks and with random frequencies. This is achieved by appealing the theory of graphons to analyze a mean-field model coming in the form of an infinite oscillator limit which provides a single master equation for studying random Kuramoto models. We show that bifurcations to synchrony and hyperbolic synchrony patterns in the mean-field model can also be found in related random Kuramoto networks for large numbers of oscillators. We further provide a detailed application of our results to oscillators arranged on Erdős--Rényi random networks, for which we further identify that not all bifurcations to synchrony emerge through simple co-dimension one bifurcations.

Figures (6)

• Figure 1: Each black dot on both figures is the location of the critical coupling value of \ref{['Kuramoto']} on an Erdős--Rényi graph with edge probability $p = 1$ (top) and $p = 0.5$ (bottom) and frequencies $\omega_j$ drawn from the uniform distribution on $[-1,1]$. For each network size $n$ there are 100 random realizations of system \ref{['Kuramoto']}, with the red line representing the mean across $n$ and the shaded region bounded by blue dashed lines denoting one standard deviation from the mean.
• Figure 2: Comparison of the synchronous solution at the critical coupling with $n = 500$ oscillators and frequencies drawn from the uniform distribution (red dots) against the continuum synchronous profile $\theta(x) = \arcsin(2x - 1)$ (black line) to \ref{['KuramotoER']}. Synchronous solutions are plotted as $\{(x_j,\theta_j)\}_{j = 1}^{500}$ with each $x_j$ drawn independently from the uniform distribution on $[0,1]$ to generate the frequencies $\omega_j = \Omega(x_j)$. Left: All-to-all coupling ($p=1$). Right: Erdős--Rényi random network $(p =0.5)$.
• Figure 3: Comparison of the synchronous solution at the critical coupling with $n = 500$ oscillators and frequencies drawn from the distribution with density \ref{['CosineDistribution']} (red dots) against the continuum synchronous profile $\theta(x) = \arcsin(\tan(\frac{\pi}{4}(2x - 1)))$ (black line) to \ref{['KuramotoER']}. Synchronous solutions are plotted as $\{(x_j,\theta_j)\}_{j = 1}^{500}$ with each $x_j$ drawn independently from the uniform distribution on $[0,1]$ to generate the frequencies $\omega_j = \Omega(x_j)$. Left: All-to-all coupling ($p=1$). Right: Erdős--Rényi random network $(p =0.5)$.
• Figure 4: Left: Continuation of synchronous solutions in $n = 500$ oscillator random Kuramoto model on an Erdős--Rényi network with $p = 0.5$ and frequencies distributed according to the density \ref{['CosineDistribution']}. Plotted is the order parameter \ref{['OrderParam']} versus the coupling coefficient $K$ with a saddle-node bifurcation leading to onset of synchronization denoted by a red dot at $K \approx 3.0328$. Linearized stability is indicated by a solid curve, while unstable solutions are along dashed curves. Right: Eigenvalues of the linearization about the synchronous solution at two points along the bifurcation curve indicated by green squares, showing a single eigenvalue cross at the bifurcation point (emphasized in green).
• Figure 5: Comparison of the synchronous solution at the critical coupling with $n = 500$ oscillators and frequencies drawn from the Cauchy distribution (red dots) against the continuum synchronous profile $\theta(x) = \arcsin(-\cos(\pi x)/1.1002)$ (black line) to \ref{['KuramotoER']}. Synchronous solutions are plotted as $\{(x_j,\theta_j)\}_{j = 1}^{500}$ with each $x_j$ drawn independently from the uniform distribution on $[0,1]$ to generate the frequencies $\omega_j = \Omega(x_j)$. Left: All-to-all coupling ($p=1$). Right: Erdős--Rényi random network $(p =0.5)$.

Theorems & Definitions (28)

• Remark 1
• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Proposition 4.1
• Lemma 4.2
• proof
• Lemma 5.1
• proof
• Lemma 5.2