Revolutionaries and Spies on Random Graphs

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 σ(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 for some ε > 0). For sparser graphs, some bounds are provided.