We study the dynamics of information (or virus) dissemination by @math mobile agents performing independent random walks on an @math -node grid. We formulate our results in terms of two scenarios: broadcasting and gossiping. In the broadcasting scenario, the mobile agents are initially placed uniformly at random among the grid nodes. At time 0, one agent is informed of a rumor and starts a random walk. When an informed agent meets an uninformed agent, the latter becomes informed and starts a new random walk. We study the broadcasting time of the system, that is, the time it takes for all agents to know the rumor. In the gossiping scenario, each agent is given a distinct rumor at time 0 and all agents start random walks. When two agents meet, they share all rumors they are aware of. We study the gossiping time of the system, that is, the time it takes for all agents to know all rumors. We prove that both the broadcasting and the gossiping times are @math w.h.p., thus achieving a tight characterization up to logarithmic factors. Previous results for the grid provided bounds which were weaker and only concerned average times. In the context of virus infection, a corollary of our results is that static and dynamically moving agents are infected at about the same speed.
With the advent of mobile ad-hoc networks there has been growing interest in studying information dissemination in dynamic scenarios, where a number of agents move either in a continuous space or along the nodes of some underlying graph and exchange information when their positions satisfy a specified proximity constraint. In @cite_15 @cite_2 the authors study the time it takes to broadcast information from one of @math mobile agents to all others. The agents move on a square grid of @math nodes and in each time step, an agent can (a) exchange information with all agents at distance at most @math from it, and (b) move to any random node at distance at most @math from its current position. The results in these papers only apply to a very dense scenario where the number of agents is linear in the number of grid nodes (i.e., @math ). They show that the broadcasting time is @math w.h.p., when @math and @math @cite_15 , and it is @math w.h.p., when @math @cite_2 . These results crucially rely on @math , which implies that the range of agents' communications or movements at each step defines a connected graph.
@cite_15 @cite_2
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Markovian evolving graphs are dynamic-graph models where the links among a fixed set of nodes change during time according to an arbitrary Markovian rule. They are extremely general and they can well describe important dynamic-network scenarios. We study the speed of information spreading in the "stationary phase" by analyzing the completion time of the "flooding mechanism". We prove a general theorem that establishes an upper bound on flooding time in any stationary Markovian evolving graph in terms of its node-expansion properties. We apply our theorem in two natural and relevant cases of such dynamic graphs. "Geometric Markovian evolving graphs" where the Markovian behaviour is yielded by "n" mobile radio stations, with fixed transmission radius, that perform independent random walks over a square region of the plane. "Edge-Markovian evolving graphs" where the probability of existence of any edge at time "t" depends on the existence (or not) of the same edge at time "t-1". In both cases, the obtained upper bounds hold "with high probability" and they are nearly tight. In fact, they turn out to be tight for a large range of the values of the input parameters. As for geometric Markovian evolving graphs, our result represents the first analytical upper bound for flooding time on a class of concrete mobile networks. We consider Mobile Ad-hoc NETworks (MANETs) formed by n nodes that move independently at random over a finite square region of the plane. Nodes exchange data if they are at distance at most r within each other, where r > 0 is the node transmission radius . The flooding time is the number of time steps required to broadcast a message from a source node to every node of the network. Flooding time is an important measure of the speed of information spreading in dynamic networks. We derive a nearly-tight upper bound on the flooding time which is a decreasing function of the maximal velocity of the nodes. It turns out that, when the node velocity is "sufficiently" high, even if the node transmission radius r is far below the connectivity threshold , the flooding time does not asymptotically depend on r . So, flooding can be very fast even though every snapshot (i.e. the static random geometric graph at any fixed time) of the MANET is fully disconnected. Our result is the first analytical evidence of the fact that high, random node mobility strongly speed-up information spreading and, at the same time, let nodes save energy .