Relationship Between the Number of Agents and Sparse Observability Index

The state estimation problem in the presence of malicious sensor attacks is commonly referred to as a secure state estimation problem. Central to addressing this problem is the concept of the sparse observability index, defined as the largest integer $ \delta$ for which the system remains observable after the removal of any $\delta$ sensors. This index plays a critical role in quantifying the resilience of the system, as a higher $\delta$ enables unique state reconstruction despite the presence of more compromised sensors. In this study, for undirected multi-agent systems consisting of $ n$ agents, we analyze the relationship between the number of agents $ n$ and the sparse observability index $ \delta$ for effective secure state estimation. In particular, we consider four typical graph structures: path, cycle, complete, and complete bipartite graphs. Our analysis reveals that $\delta$ does not increase monotonically with $n$, and that resilience is intricately tied to the underlying network structure. Notably, we demonstrate that the system exhibits enhanced resilience when the number of agents $n$ is a prime number, although the specifics of this relationship vary depending on the graph topology.