Distributed Consensus Formation Through Unconstrained Gossiping

TL;DR

This paper addresses issues in distributed consensus algorithms by introducing conflict resolution techniques and a Markov chain-based analysis, validated through simulations to improve understanding of convergence behavior.

Contribution

It proposes conflict resolution methods for gossip algorithms and develops a Markov chain framework to analyze their convergence and behavior.

Findings

Conflict resolution ensures valid consensus states.

Markov chain analysis predicts convergence probabilities and times.

Simulations validate the analytical methodology.

Abstract

Gossip algorithms are widely used to solve the distributed consensus problem, but issues can arise when nodes receive multiple signals either at the same time or before they are able to finish processing their current work load. Specifically, a node may assume a new state that represents a linear combination of all received signals; even if such a state makes no sense in the problem domain. As a solution to this problem, we introduce the notion of conflict resolution for gossip algorithms and prove that their application leads to a valid consensus state when the underlying communication network possesses certain properties. We also introduce a methodology based on absorbing Markov chains for analyzing gossip algorithms that make use of these conflict resolution algorithms. This technique allows us to calculate both the probabilities of converging to a specific consensus state and the time that such convergence is expected to take. Finally, we make use of simulation to validate our methodology and explore the temporal behavior of gossip algorithms as the size of the network, the number of states per node, and the network density increase.