Broadcast Gossip Algorithms for Consensus on Strongly Connected Digraphs

TL;DR

This paper introduces a broadcast gossip algorithm framework for achieving consensus on strongly connected directed graphs, guaranteeing convergence and average consensus under certain conditions, with improved convergence speed and reliability.

Contribution

It proposes a novel broadcast gossip algorithm with companion variables, providing convergence guarantees on strongly connected digraphs and conditions for average consensus, along with convergence rate analysis.

Findings

Guarantees convergence to consensus on strongly connected digraphs.

Ensures average consensus in expectation and mean-squared sense under certain conditions.

Achieves faster convergence with fewer broadcasts compared to existing algorithms.

Abstract

We study a general framework for broadcast gossip algorithms which use companion variables to solve the average consensus problem. Each node maintains an initial state and a companion variable. Iterative updates are performed asynchronously whereby one random node broadcasts its current state and companion variable and all other nodes receiving the broadcast update their state and companion variable. We provide conditions under which this scheme is guaranteed to converge to a consensus solution, where all nodes have the same limiting values, on any strongly connected directed graph. Under stronger conditions, which are reasonable when the underlying communication graph is undirected, we guarantee that the consensus value is equal to the average, both in expectation and in the mean-squared sense. Our analysis uses tools from non-negative matrix theory and perturbation theory. The perturbation results rely on a parameter being sufficiently small. We characterize the allowable upper bound as well as the optimal setting for the perturbation parameter as a function of the network topology, and this allows us to characterize the worst-case rate of convergence. Simulations illustrate that, in comparison to existing broadcast gossip algorithms, the approaches proposed in this paper have the advantage that they simultaneously can be guaranteed to converge to the average consensus and they converge in a small number of broadcasts.