Gossip over Holonomic Graphs

TL;DR

This paper introduces a novel holonomy concept to establish order-independent convergence conditions for gossip processes on graphs, broadening understanding of multi-agent consensus dynamics.

Contribution

It provides a necessary and sufficient condition for convergence of gossip processes that does not depend on iteration order, using the new notion of holonomy.

Findings

Convergence condition independent of iteration order

Characterization of the limit states of the process

Introduction of holonomy for local stochastic matrices

Abstract

A gossip process is an iterative process in a multi-agent system where only two neighboring agents communicate at each iteration and update their states. The neighboring condition is by convention described by an undirected graph. In this paper, we consider a general update rule whereby each agent takes an arbitrary weighted average of its and its neighbor's current states. In general, the limit of the gossip process (if it converges) depends on the order of iterations of the gossiping pairs. The main contribution of the paper is to provide a necessary and sufficient condition for convergence of the gossip process that is independent of the order of iterations. This result relies on the introduction of the novel notion of holonomy of local stochastic matrices for the communication graph. We also provide complete characterizations of the limit and the space of holonomic stochastic matrices over the graph.