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^{1/2+\eps} for some
\eps > 0). For sparser graphs, some bounds are provided