Scalable Resetting Algorithms for Synchronization of Pulse-Coupled Oscillators over Rooted Directed Graphs

TL;DR

This paper introduces scalable resetting algorithms for pulse-coupled oscillators on directed graphs, achieving robust, fixed-time synchronization in rooted acyclic graphs and almost sure synchronization in all rooted digraphs using stochastic modifications.

Contribution

The paper presents new scalable deterministic and stochastic resetting algorithms for PCO synchronization on directed graphs, overcoming previous scalability limitations and establishing impossibility results.

Findings

Deterministic algorithms achieve fixed-time synchronization in rooted acyclic digraphs.

Stochastic algorithms guarantee almost sure synchronization in all rooted digraphs.

Algorithms' parameters are network independent, enhancing scalability.

Abstract

We study the problem of robust global synchronization of pulse-coupled oscillators (PCOs) over directed graphs. It is known that when the digraphs are strongly connected, global synchronization can be achieved by using a class of deterministic set-valued reset controllers. However, for large-scale networks, these algorithms are not scalable because some of their tuning parameters have upper bounds of the order of O(1/N), where N is the number of agents. This paper resolves this scalability issue by presenting several new results in the context of global synchronization of PCOs with more general network topologies using deterministic and stochastic hybrid dynamical systems. First, we establish that similar deterministic resetting algorithms can achieve robust, global, and fixed-time synchronization in any rooted acyclic digraph. Moreover, in this case we show that the synchronization dynamics are now scalable as the tuning parameters of the algorithm are network independent, i.e., of order O(1). However, the algorithms cannot be further extended to all rooted digraphs. We establish this new impossibility result by introducing a counterexample with a particular rooted digraph for which global synchronization cannot be achieved, irrespective of the tuning of the reset rule. Nevertheless, we show that if the resetting algorithms are modified by accommodating an Erdos-Renyi type random graph model, then the resulting stochastic resetting dynamics will guarantee global synchronization almost surely for all rooted digraphs and, moreover, the tunable parameters of the dynamics are network independent. Stability and robustness properties of the resetting algorithms are studied using the tools from set-valued hybrid dynamical systems. Numerical simulations are provided at the end of the paper for demonstration of the main results.