Minimal Construction of Graphs with Maximum Robustness

TL;DR

This paper identifies the minimal edge structures needed for undirected graphs to achieve maximum robustness, enabling resilient network consensus with fewer communication resources.

Contribution

It establishes tight necessary conditions for minimal edges in robust graphs and constructs classes of graphs that meet maximum robustness with minimal edges.

Findings

Derived tight necessary edge count conditions for maximum robustness.

Constructed minimal edge robust graphs (MERGs) achieving maximum robustness.

Validated effectiveness through comparisons and simulations.

Abstract

The notions of $r$-robustness and $(r,s)$-robustness of a network have been earlier introduced in the literature to achieve resilient consensus in the presence of misbehaving agents. However, while higher robustness levels enable networks to tolerate a higher number of misbehaving agents, they also require dense communication structures, which are not always desirable for systems with limited communication ranges, energy, and resources. Therefore, this paper studies the fundamental structures behind $r$-robustness and $(r,s)$- robustness properties in two ways. (a) We first establish tight necessary conditions on the number of edges that an undirected graph with an arbitrary number of nodes must have to achieve maximum $r$- and $(r,s)$-robustness. (b) We then use these conditions to construct two classes of undirected graphs, referred as to $\gamma$- and $(\gamma,\gamma)$-Minimal Edge Robust Graphs (MERGs), that provably achieve maximum robustness with minimal numbers of edges. We demonstrate the effectiveness of our method via comparison against existing robust graph structures and a set of simulations.