Fast Consensus over Almost Regular Directed Graphs

TL;DR

This paper investigates the design of directed graphs that optimize algebraic connectivity to achieve fast consensus, providing theoretical characterizations and efficient constructions for various graph densities.

Contribution

It identifies optimal directed graphs for maximizing algebraic connectivity and proposes a computational method for near-optimal graph sequences.

Findings

Optimal directed graphs for sparse and dense cases are characterized.

A sequence of almost regular graphs achieves near-optimal algebraic connectivity.

The proposed methods lead to faster consensus in directed networks.

Abstract

This paper studies an open consensus network design problem: identifying the optimal simple directed graphs, given a fixed number of vertices and arcs, that maximize the second smallest real part of all Laplacian eigenvalues, referred to as algebraic connectivity. For sparse and dense graphs, the class of all optimal directed graphs that maximize algebraic connectivity is theoretically identified, leading to the fastest consensus. For general graphs, a computationally efficient sequence of almost regular directed graphs is proposed to achieve fast consensus, with algebraic connectivity close to the optimal value.