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 @math mobile agents performing independent random walks on an @math -node grid. At time @math , 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 @math , where two agents are connected by an edge in @math iff their distance at time @math is within their transmission radius @math . 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 @math before the graph is altered by the motion. We study the broadcast time @math 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 @math ) where, with high probability, no connected component in @math 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 each other. Quite surprisingly, we show that for a system below the percolation point the broadcast time does not depend on the relation between the mobility speed and the transmission radius. In fact, we prove that @math for any @math , even when the transmission range is significantly larger than the mobility range in one step, giving a tight characterization up to logarithmic factors. Our result complements a recent result of (SODA 2011) who showed that above the percolation point the broadcast time is polylogarithmic in @math .
@cite_21 @cite_24 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 broadcast time is @math w.h.p., when @math and @math @cite_21 , and it is @math w.h.p., when @math @cite_24 . These results crucially rely on @math , which implies that the range of agents' communications or movements at each step defines a connected graph.
@cite_24 @cite_21
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We consider a Mobile Ad-hoc NETworks (MANET) 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". This implies that 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. Data reach all nodes quickly despite these ones use very low transmission power. 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. 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.