Reaching Approximate Byzantine Consensus with Multi-hop Communication

TL;DR

This paper investigates the conditions under which approximate Byzantine consensus can be achieved in networks with multi-hop communication, providing necessary and sufficient topological criteria and a new algorithm based on a minimal message cover property.

Contribution

It introduces a comprehensive analysis of multi-hop relay effects on Byzantine consensus, extending existing conditions for local communication to broader multi-hop scenarios and proposing a novel algorithm.

Findings

Conditions for consensus are weaker for multi-hop communication with l>1.

For undirected graphs, n≥3f+1 and node connectivity ≥2f+1 are necessary and sufficient.

The proposed trim function algorithm works for multi-graphs and is based on a minimal message cover property.

Abstract

We address the problem of reaching consensus in the presence of Byzantine faults. In particular, we are interested in investigating the impact of messages relay on the network connectivity for a correct iterative approximate Byzantine consensus algorithm to exist. The network is modeled by a simple directed graph. We assume a node can send messages to another node that is up to $l$ hops away via forwarding by the intermediate nodes on the routes, where $l\in \mathbb{N}$ is a natural number. We characterize the necessary and sufficient topological conditions on the network structure. The tight conditions we found are consistent with the tight conditions identified for $l=1$, where only local communication is allowed, and are strictly weaker for $l>1$. Let $l^*$ denote the length of a longest path in the given network. For $l\ge l^*$ and undirected graphs, our conditions hold if and only if $n\ge 3f+1$ and the node-connectivity of the given graph is at least $2f+1$ , where $n$ is the total number of nodes and $f$ is the maximal number of Byzantine nodes; and for $l\ge l^*$ and directed graphs, our conditions is equivalent to the tight condition found for exact Byzantine consensus. Our sufficiency is shown by constructing a correct algorithm, wherein the trim function is constructed based on investigating a newly introduced minimal messages cover property. The trim function proposed also works over multi-graphs.