A Sample-Based Algorithm for Approximately Testing r-Robustness of a Digraph

One of the intensely studied concepts of network robustness is r-robustness, which is a network topology property quantified by an integer r. It is required by mean subsequence reduced (MSR) algorithms and their variants to achieve resilient consensus. However, determining r-robustness is intractable for large networks. In this paper, we propose a sample-based algorithm to approximately test r-robustness of a digraph with n vertices and m edges. For a digraph with a moderate assumption on the minimum in-degree, and an error parameter 0 <ϵ ≤ 1, the proposed algorithm distinguishes (r+ ϵn)-robust graphs from graphs which are not r-robust with probability (1−δ). Our algorithm runs in $\exp \left( {O\left( {\left( {\ln \frac{1}{{ \in \delta }}} \right)/{ \in ^2}} \right)} \right) \cdot m$ time. The running time is linear in the number of edges if ϵ is a constant.