Global synchronization of pulse-coupled oscillators on trees

TL;DR

This paper introduces a new inhibitory pulse-coupling method for phase oscillators on trees, proving synchronization within a bounded time and developing a universal randomized clock synchronization algorithm with efficient convergence properties.

Contribution

It presents a novel inhibitory pulse-coupling technique and an adaptive extension that guarantees synchronization on trees, along with a universal randomized clock synchronization algorithm.

Findings

Synchronization achieved within 51d time for trees with degree ≤ 3.

Adaptive pulse-coupling synchronizes arbitrary initial states within 83d.

Universal clock synchronization algorithm converges in expected O(|V|+(d^5+Δ^2)log|V|) time.

Abstract

Consider a distributed network on a finite simple graph $G=(V,E)$ with diameter $d$ and maximum degree $\Delta$, where each node has a phase oscillator revolving on $S^{1}=\mathbb{R}/\mathbb{Z}$ with unit speed. Pulse-coupling is a class of distributed time evolution rule for such networked phase oscillators inspired by biological oscillators, which depends only upon event-triggered local pulse communications. In this paper, we propose a novel inhibitory pulse-coupling and prove that arbitrary phase configuration on $G$ synchronizes by time $51d$ if $G$ is a tree and $\Delta \le 3$. We extend this pulse-coupling by letting each oscillator throttle the input according to an auxiliary state variable. We show that the resulting adaptive pulse-coupling synchronizes arbitrary initial configuration on $G$ by time $83d$ if $G$ is a tree. As an application, we obtain a universal randomized distributed clock synchronization algorithm, which uses $O(\log \Delta)$ memory per node and converges on any $G$ with expected worst case running time of $O(|V|+(d^{5}+\Delta^{2})\log |V|)$.