  We address the question of understanding the effect of the underlying network
topology on the spread of a virus and the dissemination of information when
users are mobile performing independent random walks on a graph. To this end we
propose a simple model of infection that enables to study the coincidence time
of two random walkers on an arbitrary graph. By studying the coincidence time
of a susceptible and an infected individual both moving in the graph we obtain
estimates of the infection probability. The main result of this paper is to
pinpoint the impact of the network topology on the infection probability. More
precisely, we prove that for homogeneous graph including regular graphs and the
classical Erdos-Renyi model, the coincidence time is inversely proportional to
the number of nodes in the graph. We then study the model on power-law graphs,
that exhibit heterogeneous connectivity patterns, and show the existence of a
phase transition for the coincidence time depending on the parameter of the
power-law of the degree distribution.
