Secure State Estimation for Multi-Agent Systems: On the Relationship between the Number of Agents and System Resilience

This paper addresses the problem of secure state estimation in multi-agent systems consisting of n agents. For the problem, the sparse observability index is a well-known measure of the system resilience. This index is defined as the largest integer δ for which the system observability is preserved even if any δ sensors are eliminated. The larger δ is, the more the system state can be reconstructed even if the system is subjected to more sensor attacks. In this paper, we analyze the relationship between the number of agents n and the sparse observability index δ especially when the network structure is path and cycle graphs. Intuitively, it is assumed that, as the number of agents n increases, the system resilience, i.e., δ, increases accordingly. However, as this study provides, δ does not monotonically increase with an increase in n. In path graphs, δ depends on whether n+1 is prime or composite, and the system resilience is enhanced when n+1 is prime. In cycle graphs, δ obeys whether n is prime or composite, and the system is more resilient when n is prime.