Construction of the Sparsest Maximally r-Robust Graphs

Abstract

In recent years, the notion of r-robustness for the communication graph of the network has been introduced to address the challenge of achieving consensus in the presence of misbehaving agents. Higher r-robustness typically implies higher tolerance to malicious information towards achieving resilient consensus, but it also implies more edges for the communication graph. This in turn conflicts with the need to minimize communication due to limited resources in real-world applications (e.g., multi-robot networks). In this paper, our contributions are twofold. (a) We provide the subgraph structures and tight lower bounds on the number of edges required for graphs with a given number of nodes to reach the maximum robustness. (b) We then use the results of (a) to introduce two classes of graphs that utilize the least number of edges to maintain maximum robustness. Our work is validated through a series of simulations.