Pursuit-Evasion in Models of Complex Networks

Abstract

Pursuit-evasion games, such as the game of Cops and Robbers, are a simplified model for network security. In this game, cops try to capture a robber loose on the vertices of the network. The minimum number of cops required to win on a graph $G$ is its number. We present asymptotic results for the game of Cops and Robbers played in various stochastic network models, such as in $G(n,p)$ with nonconstant $p$ and in random power-law graphs. We find bounds for the cop number of $G(n,p)$ for a large range $p$ as a function of $n$. We prove that the cop number of random power-law graphs with $n$ vertices is asymptotically almost surely $\Theta(n)$. The cop number of the core of random power-law graphs is investigated, and it is proved to be of smaller order than the order of the core.