Fastest Distributed Consensus Averaging Problem on Chain of Rhombus Networks

TL;DR

This paper derives an analytical solution for the fastest distributed consensus averaging algorithm on Chain of Rhombus networks, improving convergence rates by leveraging graph stratification and semidefinite programming.

Contribution

It introduces a novel analytical method for optimizing consensus algorithms specifically on Chain of Rhombus networks, utilizing stratification and slackness conditions.

Findings

Optimal weights are derived through characteristic polynomial analysis.

Replacing nodes with rhombus subgraphs enhances convergence rate.

The eigenvalues and SLEM of the network are explicitly characterized.

Abstract

Distributed consensus has appeared as one of the most important and primary problems in the context of distributed computation and it has received renewed interest in the field of sensor networks (due to recent advances in wireless communications), where solving fastest distributed consensus averaging problem over networks with different topologies is one of the primary problems in this issue. Here in this work analytical solution for the problem of fastest distributed consensus averaging algorithm over Chain of Rhombus networks is provided, where the solution procedure consists of stratification of associated connectivity graph of the network and semidefinite programming, particularly solving the slackness conditions, where the optimal weights are obtained by inductive comparing of the characteristic polynomials initiated by slackness conditions. Also characteristic polynomial together with its roots corresponding to eigenvalues of weight matrix including SLEM of network is determined inductively. Moreover to see the importance of rhombus graphs it is indicated that convergence rate of path network increases by replacing a single node by a rhombus sub graph within the path network.