# Convergence Results on Pulse Coupled Oscillator Protocols in Locally Connected Networks

## TL;DR

This paper analyzes the convergence properties of pulse coupled oscillator networks in locally connected topologies, providing new theoretical insights into synchronization and desynchronization behaviors relevant for decentralized communication systems.

## Contribution

It offers the first comprehensive convergence analysis of PCOs in locally connected networks, including accuracy bounds and conditions for synchronization and scheduling.

## Key findings

- Proves almost sure convergence for 3-node networks.
- Derives accuracy results for general locally connected networks.
- Establishes convergence conditions for desynchronization protocols.

## Abstract

This work provides new insights on the convergence of a locally connected network of pulse coupled oscillator (PCOs) (i.e., a bio-inspired model for communication networks) to synchronous and desynchronous states, and their implication in terms of the decentralized synchronization and scheduling in communication networks. Bio-inspired techniques have been advocated by many as fault-tolerant and scalable alternatives to produce self-organization in communication networks. The PCO dynamics in particular have been the source of inspiration for many network synchronization and scheduling protocols. However, their convergence properties, especially in locally connected networks, have not been fully understood, prohibiting the migration into mainstream standards. This work provides further results on the convergence of PCOs in locally connected networks and the achievable convergence accuracy under propagation delays. For synchronization, almost sure convergence is proved for $3$ nodes and accuracy results are obtained for general locally connected networks whereas, for scheduling (or desynchronization), results are derived for locally connected networks with mild conditions on the overlapping set of maximal cliques. These issues have not been fully addressed before in the literature.