r-Robustness and (r,s)-Robustness of Circulant Graphs

TL;DR

This paper investigates the r- and (r,s)-robustness properties of a specific class of scalable directed circulant graphs, providing theoretical results and simulations to understand their robustness levels.

Contribution

It introduces a class of directed circulant graphs with robustness determined by a parameter, extending robustness analysis to directed graphs.

Findings

Robustness of the graphs is determined by the parameter k.

Theoretical proofs establish the robustness levels.

Simulations support the theoretical results.

Abstract

There has been recent growing interest in graph theoretical properties known as r- and (r,s)-robustness. These properties serve as sufficient conditions guaranteeing the success of certain consensus algorithms in networks with misbehaving agents present. Due to the complexity of determining the robustness for an arbitrary graph, several methods have previously been proposed for identifying the robustness of specific classes of graphs or constructing graphs with specified robustness levels. The majority of such approaches have focused on undirected graphs. In this paper we identify a class of scalable directed graphs whose edge set is determined by a parameter k and prove that the robustness of these graphs is also determined by k. We support our results through computer simulations.