  Dynamical processes taking place on networks have received much attention in
recent years, especially on various models of random graphs (including small
world and scale free networks). They model a variety of phenomena, including
the spread of information on the Internet; the outbreak of epidemics in a
spatially structured population; and communication between randomly dispersed
processors in an ad hoc wireless network. Typically, research has concentrated
on the existence and size of a large connected component (representing, say,
the size of the epidemic) in a percolation model, or uses differential
equations to study the dynamics using a mean-field approximation in an infinite
graph. Here we investigate the time taken for information to propagate from a
single source through a finite network, as a function of the number of nodes
and the network topology. We assume that time is discrete, and that nodes
attempt to transmit to their neighbors in parallel, with a given probability of
success. We solve this problem exactly for several specific topologies, and use
a large-deviation theorem to derive general asymptotic bounds, which apply to
any family of networks where the diameter grows at least logarithmically in the
number of nodes. We use these bounds, for example, to show that a scale-free
network has propagation time logarithmic in the number of nodes, and inversely
proportional to the transmission probability.
