Consensus over Weighted Directed Graphs: A Robustness Perspective

TL;DR

This paper analyzes the robustness of consensus protocols on weighted directed graphs, deriving explicit limits on weight variations for consensus achievement, including cases with negative weights, supported by theoretical and simulation results.

Contribution

It provides explicit bounds on weight variations for consensus in directed graphs, including negative weights, and offers graph-theoretic interpretations for specific graph structures.

Findings

Consensus can be achieved despite negative weights.

Explicit limits on weight variation are derived.

Simulations confirm theoretical predictions.

Abstract

The present paper investigates the robustness of the consensus protocol over weighted directed graphs using the Nyquist criterion. The limit to which a single weight can vary, while consensus among the agents can be achieved, is explicitly derived. It is shown that even with a negative weight on one of the edges, consensus may be achieved. The result obtained in this paper is applied to a directed acyclic graph and to the directed cycle graph. Graph theoretic interpretations of the limits are provided for the two cases. Simulations support the theoretical results.