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 .
A model similar to our scenario is often employed to represent the spreading of computer viruses in networks and the spreading time is also referred to as . Kesten and Sidoravicius @cite_9 characterized the rate at which an infection spreads among particles performing continuous-time random walks with the same jump rate. In @cite_6 , the authors provide a general bound on the average infection time when @math agents (one of them initially affected by the virus) move in an @math -node graph. For general graphs, this bound is @math , where @math denotes the maximum average meeting time of two random walks on the graph, and the maximum is taken over all pairs of starting locations of the random walks. Also, in the paper tighter bounds are provided for the complete graph and for expanders. Observe that the @math bound specializes to @math for the @math -node grid by applying the known bound on @math of @cite_18 . A tight bound of @math on the infection time on the grid is claimed in @cite_0 , based on a rather informal argument where some unwarranted independence assumptions are made. Our results show that this latter bound is incorrect.
@cite_0 @cite_9 @cite_18 @cite_6
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We study the process of the spread of an infection among mobile nodes moving on a finite, grid based map. A random walk and a novel adversarial model are considered as two extreme cases of node mobility. With N nodes, we present analytical and simulation results for both mobility models for a square grid map with size √G × √G. A key finding is that with random mobility the total time to infect all nodes decreases with N while with an adversarial model we observe a reverse trend. Specifically, the random case results in a total infection time of Θ(GlogGlogN (N) as opposed to the adversarial case where the total infection time is found to be Θ(√(Glog(N). We also explore the possibility of emulating such an infection process as a mobile interaction game with wireless sensor motes, and the above results are complimented by traces obtained from an empirical study with humans as players in an outdoor field. In the models we will consider, space is represented by a grid of sites that can be in one of a finite number of states and that change at rates that depend on the states of a finite number of sites. Our main aim here is to explain an idea of Durrett and Levin (1994): the behavior of these models can be predicted from the properties of the mean field ODE, i.e., the equations for the densities of the various types that result from pretending that all sites are always independent. We will illustrate this picture through a discussion of eight families of examples from statistical mechanics, genetics, population biology, epidemiology, and ecology. Some of our findings are only conjectures based on simulation, but in a number of cases we are able to prove results for systems with "fast stirring" by exploiting connections between the spatial model and an associated reaction diffusion equation. Consider k particles, 1 red and k - 1 white, chasing each other on the nodes of a graph G. If the red one catches one of the white, it "infects" it with its color. The newly red particles are now available to infect more white ones. When is it the case that all white will become red? It turns out that this simple question is an instance of information propagation between random walks and has important applications to mobile computing where a set of mobile hosts acts as an intermediary for the spread of information. In this paper we model this problem by k concurrent random walks, one corresponding to the red particle and k - 1 to the white ones. The infection time Tk of infecting all the white particles with red color is then a random variable that depends on k, the initial position of the particles, the number of nodes and edges of the graph, as well as on the structure of the graph. In this work we develop a set of probabilistic tools that we use to obtain upper bounds on the (worst case w.r.t. initial positions of particles) expected value of Tk for general graphs and important special cases. We easily get that an upper bound on the expected value of Tk is the worst case (over all initial positions) expected meeting time m* of two random walks multiplied by Θ(log k). We demonstrate that this is, indeed, a tight bound; i.e. there is a graph G (a special case of the "lollipop" graph), a range of values k < n (such that √n - k = Θ(√n)) and an initial position of particles achieving this bound. When G is a clique or has nice expansion properties, we prove much smaller bounds for Tk. We have evaluated and validated all our results by large scale experiments which we also present and discuss here. In particular, the experiments demonstrate that our analytical results for these expander graphs are tight.