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

TL;DR

This paper introduces a probabilistic, sample-based algorithm to efficiently approximate the $r$-robustness of directed graphs, enabling analysis of large networks where exact computation is infeasible.

Contribution

The paper presents a novel sampling method that approximates $r$-robustness with probabilistic guarantees, reducing computational complexity for large networks.

Findings

Algorithm distinguishes $(r+ ext{error})$-robust graphs with high probability.

Runtime is exponential in inverse squared error but linear in edges for fixed error.

Effective for large networks with moderate in-degree assumptions.

Abstract

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 $0<\epsilon\leq 1$, the proposed algorithm distinguishes $(r+\epsilon n)$-robust graphs from graphs which are not $r$-robust with probability $(1-\delta)$. Our algorithm runs in $\exp(O((\ln{\frac{1}{\epsilon\delta}})/\epsilon^2))\cdot m$ time. The running time is linear in the number of edges if $\epsilon$ is a constant.