We introduce stochastic time-dependency in evolving graphs: starting from in arbitrary, initial edge probability distribution, at every time step! every edge changes it's state (existing or not) according to a two-state Markovian process with probabilities 1) (edge birth-rate) and q (edge death-rate). If all edge exists at time t then, at time t+1 it dies with probability q. If instead the edge does not exist at time 1, then it will come into existence at time t + 1 with Probability 1). Such evolving graph model is a. wide generalization of time-independent dynamic random graphs [6] and will be called edge-Markovian. dynamic graphs. We investigate the speed of information dissemination in such dynamic graphs. We provide nearly tight; bounds (which in fact turn out to be tight for a wide range of probabilities p and q) oil the completion Chile of the flooding mechanism aiming to broadcast a piece of information from a source node to all nodes. In particular, we provide: i) A tight characterization of the class of edge-Markovian dynamic graphs where flooding time is constant and. thus, it does not asymptotically depend oil the initial probability distribution. ii) A flight characterization of the class of edge-Markovian dynamic graphs where flooding time does not asymptotically depend oil the edge death-rate q

We introduce stochastic time-dependency in evolving graphs: starting from an initial graph, at every time step, every edge changes its state (existing or not) according to a two-state Markovian process with probabilities $p$ (edge birth-rate) and $q$ (edge death-rate). If an edge exists at time $t$, then, at time $t+1$, it dies with probability $q$. If instead the edge does not exist at time $t$, then it will come into existence at time $t+1$ with probability $p$. Such an evolving graph model is a wide generalization of time-independent dynamic random graphs [A. E. F. Clementi, A. Monti, F. Pasquale, and R. Silvestri, J. Comput. System Sci., 75 (2009), pp. 213–220] and will be called edge-Markovian evolving graphs. We investigate the speed of information spreading in such evolving graphs. We provide nearly tight bounds (which in fact turn out to be tight for a wide range of probabilities $p$ and $q$) on the completion time of the flooding mechanism aiming to broadcast a piece of information from a source node to all nodes. In particular, we provide i) a tight characterization of the class of edge-Markovian evolving graphs where flooding time is constant and, thus, it does not asymptotically depend on the initial graph; ii) a tight characterization of the class of edge-Markovian evolving graphs where flooding time does not asymptotically depend on the edge death-rate $q$. An interesting consequence of our results is that information spreading can be fast even if the graph, at every time step, is very sparse and disconnected. Furthermore, our bounds imply that the flooding time can be exponentially shorter than the mixing time of the edge-Markovian graph

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We prove large deviation principles for two versions of fractional Poisson processes: the main version is a renewal process, the alternative version is a weighted Poisson process. We also present asymptotic results for the ruin probabilities of an insurance model with a fractional Poisson claim number process. (C) 2013 Elsevier B.V. All rights reserved