Tight Bounds on Information Dissemination in Sparse Mobile Networks

Motivated by the growing interest in mobile systems, we study the dynamics of information dissemination between agents moving independently on a plane. Formally, we consider k mobile agents performing independent random walks on an n-node grid. At time 0, each agent is located at a random node of the grid and one agent has a rumor. The spread of the rumor is governed by a dynamic communication graph process {G_t(r) | t >=0}, where two agents are connected by an edge in G_t(r) if their distance at time t is within their transmission radius r. Modeling the physical reality that the speed of radio transmission is much faster than the motion of the agents, we assume that the rumor can travel throughout a connected component of G_t before the graph is altered by the motion. We study the broad- cast time T_B of the system, which is the time it takes for all agents to know the rumor. We focus on the sparse case (below the percolation point r_c ~ sqrt(n/k) ) where, with high probability, no connected component in Gt has more than a logarithmic number of agents and the broadcast time is dominated by the time it takes for many independent random walks to meet one other. Quite surprisingly, we show that for a system below the percolation point, the broadcast time does not depend on the transmission radius. In fact, we prove that TB = Theta~(n/sqrt(k)) for any 0 <= r < r_c, even when the transmission range is signicantly larger than the mobility range in one step, giving a tight characterization up to logarithmic factors. Our result complements a recent result of Peres et al. (SODA 2011) who showed that above the percolation point the broadcast time is polylogarithmic in k.