Francesco Pasquale

Flooding time in edge-Markovian dynamic graphs

We introduce stochastic time-dependency in evolving graphs: starting from an arbitrary initial edge probability distribution, 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 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) 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 dynamic graphs where flooding time is constant and, thus, it does not asymptotically depend on the initial probability distribution. ii) A tight characterization of the class of edge-Markovian dynamic graphs where flooding time does not asymptotically depend on the edge death-rate q.

Information spreading in stationary Markovian evolving graphs

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.

Flooding Time of Edge-Markovian Evolving Graphs

=1We 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

Information Spreading in Stationary Markovian Evolving Graphs

p

Broadcasting in dynamic radio networks

(edge birth-rate) and

A Note on Uniform Power Connectivity in the SINR Model

q

Communication in dynamic radio networks

(edge death-rate). If an edge exists at time

Opportunistic MANETs: mobility can make up for low transmission power

t

Plurality consensus in the gossip model

, then, at time

Metastability of logit dynamics for coordination games

t+1