The infection time of graphs
Consider k  particles, 1 red and k-1 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 k - 1 to the white ones. The infection time Tk T k 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 T k for general graphs and important special cases. We easily get that an upper bound on the expected value of Tk T k is the worst case (over all initial positions) expected meeting time m* m * of two random walks multiplied by ? ( 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 k < n (such that n - k = ? ( 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 T k . 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.
infection time random walk markov chains