Percolation critical exponents in cluster kinetics of pulse-coupled oscillators

TL;DR

This paper analyzes the transient dynamics of pulse-coupled oscillators using a percolation model, deriving critical exponents and revealing a discontinuous giant cluster formation at the percolation threshold.

Contribution

It introduces a percolation-based framework to describe cluster size evolution in pulse-coupled oscillators, deriving critical exponents and explaining the discontinuous formation of a giant cluster.

Findings

Cluster size distribution follows a specific scaling form near the percolation threshold.

A giant cluster forms discontinuously at the critical point.

Further aggregation beyond the threshold is suppressed, unlike in standard 1D percolation.

Abstract

Transient dynamics leading to the synchrony of pulse-coupled oscillators has previously been studied as an aggregation process of synchronous clusters, and a rate equation for the cluster size distribution has been proposed. However, the evolution of the cluster size distribution for general cluster sizes has not been solved yet. In this paper, we study the evolution of the cluster size distribution from the perspective of a percolation model by regarding the number of aggregations as the number of attached bonds. Specifically, we derive the scaling form of the cluster size distribution with specific values of the critical exponents using the property that the characteristic cluster size diverges as the percolation threshold is approached from below. Through simulation, it is confirmed that the scaling form well explains the evolution of the cluster size distribution. Based on the distribution behavior, we find that a giant cluster of all oscillators is formed discontinuously at the threshold and also that further aggregation does not occur like in a one-dimensional bond percolation model. Finally, we discuss the origin of the discontinuous formation of the giant cluster from the perspective of global suppression in explosive percolation models. For this, we approximate the aggregation process as a cluster--cluster aggregation with a given collision kernel. We believe that the theoretical approach presented in this paper can be used to understand the transient dynamics of a broad range of synchronizations.