Performance bound analysis of linear consensus algorithm on strongly connected graphs using effective resistance and reversiblization

TL;DR

This paper derives bounds on the performance of linear consensus algorithms on directed graphs by extending analysis techniques to nonreversible cases using effective resistance and new concepts.

Contribution

It introduces novel methods and concepts to analyze nonreversible directed graphs, extending previous reversible-focused results to a broader setting.

Findings

Derived bounds on LQ cost for directed graphs

Extended analysis to nonreversible cases using new concepts

Applied framework to Cayley and geometric graphs

Abstract

We study the performance of the linear consensus algorithm on strongly connected directed graphs using the linear quadratic (LQ) cost as a performance measure. In particular, we derive bounds on the LQ cost by leveraging effective resistance and reversiblization. Our results extend previous analyses-which were limited to reversible cases-to the nonreversible setting. To facilitate this generalization, we introduce novel concepts, termed the back-and-forth path and the pivot node, which serve as effective alternatives to traditional techniques that require reversibility. Moreover, we apply our approach to Cayley graphs and random geometric graphs to estimate the LQ cost without the reversibility assumption. The proposed approach provides a framework that can be adapted to other contexts where reversibility is typically assumed.