Revolutionaries and spies on random graphs

TL;DR

This paper analyzes the game of Revolutionaries and Spies on random graphs, providing asymptotic results for the minimum number of spies needed to prevent revolutionaries from holding an unguarded meeting, especially in dense graphs.

Contribution

It offers the first asymptotic analysis of the spy number on random graphs, particularly for dense graphs with high average degree.

Findings

Complete characterization of the spy number for dense random graphs.

Bounds for the spy number in sparser graphs.

Asymptotic behavior of the game parameters on $G(n,p)$.

Abstract

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of $r$ revolutionaries tries to hold an unguarded meeting consisting of $m$ revolutionaries. A team of $s$ spies wants to prevent this forever. For given $r$ and $m$, the minimum number of spies required to win on a graph $G$ is the spy number $\sigma(G,r,m)$. We present asymptotic results for the game played on random graphs $G(n,p)$ for a large range of $p = p(n), r=r(n)$, and $m=m(n)$. The behaviour of the spy number is analyzed completely for dense graphs (that is, graphs with average degree at least $n^{1/2+\eps}$ for some $\eps > 0$). For sparser graphs, some bounds are provided.