On the non-resiliency of subsequence reduced resilient consensus in multiagent networks

This paper studies resilient distributed consensus in networks lacking the structural robustness necessary for achieving consensus in the presence of misbehaving agents. Existing resilient consensus solutions, including widely adapted weighted mean subsequence reduced (WMSR) resilient consensus algorithm, present robustness conditions guaranteeing consensus among normal agents. However, when the graph is less robust than required, they only inform that agents fail to achieve consensus and do not evaluate the network performance comprehensively in such non-ideal scenarios. To address this limitation, we analyze the performance of resilient consensus in non-ideal situations by introducing the concept of non-convergent nodes. These nodes/agents cannot achieve consensus with any arbitrary agent due to the presence of misbehaving agents in the network. This notion enables ordering graphs that lack required robustness and facilitates the assessment of partial performance. Additionally, we demonstrate that among graphs with the same level of robustness (measured by their $(r,s)$-robustness), the number of non-convergent nodes varies significantly, indicating differing degrees of non-resilience. We also present numerical evaluation of results. Our approach quantifies the network performance under sub-optimal robustness conditions and offers a comprehensive resilience perspective.