An Upper Bound on the Convergence Time for Quantized Consensus of Arbitrary Static Graphs

TL;DR

This paper establishes an upper bound of O(N^3 log N) on the expected convergence time for quantized consensus algorithms in arbitrary static networks, improving previous bounds and not depending on graph topology.

Contribution

The paper derives a new, tighter upper bound on convergence time for quantized consensus in arbitrary graphs using electric network theory and Markov chain couplings.

Findings

Expected convergence time is bounded by O(N^3 log N)

Bound is independent of graph topology

Analysis extends to specific graph types like complete graphs

Abstract

We analyze a class of distributed quantized consensus algorithms for arbitrary static networks. In the initial setting, each node in the network has an integer value. Nodes exchange their current estimate of the mean value in the network, and then update their estimation by communicating with their neighbors in a limited capacity channel in an asynchronous clock setting. Eventually, all nodes reach consensus with quantized precision. We analyze the expected convergence time for the general quantized consensus algorithm proposed by Kashyap et al \cite{Kashyap}. We use the theory of electric networks, random walks, and couplings of Markov chains to derive an $O(N^3\log N)$ upper bound for the expected convergence time on an arbitrary graph of size $N$, improving on the state of art bound of $O(N^5)$ for quantized consensus algorithms. Our result is not dependent on graph topology. Example of complete graphs is given to show how to extend the analysis to graphs of given topology.