Fastest Distributed Consensus Averaging Problem on Perfect and Complete n-ary Tree networks

TL;DR

This paper provides an analytical solution for the fastest distributed consensus averaging problem on perfect and complete n-ary tree networks, optimizing edge weights to minimize convergence time.

Contribution

It introduces a convexity-based method using stratification and semidefinite programming to find optimal weights for specific tree topologies, independently of the entire network.

Findings

Derived explicit optimal weights for perfect and complete n-ary trees.

Demonstrated the convexity of the consensus averaging problem.

Provided a systematic approach for weight optimization in tree networks.

Abstract

Solving fastest distributed consensus averaging problem (i.e., finding weights on the edges to minimize the second-largest eigenvalue modulus of the weight matrix) over networks with different topologies is one of the primary areas of research in the field of sensor networks and one of the well known networks in this issue is tree network. Here in this work we present analytical solution for the problem of fastest distributed consensus averaging algorithm by means of stratification and semidefinite programming, for two particular types of tree networks, namely perfect and complete n-ary tree networks. Our method in this paper is based on convexity of fastest distributed consensus averaging problem, and inductive comparing of the characteristic polynomials initiated by slackness conditions in order to find the optimal weights. Also the optimal weights for the edges of certain types of branches such as perfect and complete n-ary tree branches are determined independently of rest of the network.