On the Speed of Information Spreading in Dynamic Networks

There is a growing interest in the study of graphs that evolve over time. Communication networks, peer-to-peer systems, social networks, ad-hoc radio networks and the Internet are only few examples of networks that are intrinsically dynamic. In order to analyse real world phenomena from a mathematical point of view, we have to define abstract models of concrete scenarios. This introduces a natural trade-off: the more a model is realistic, the more it is hard to analyse. Finding the right balance between these two poles is one of the hardest part of the work. In the thesis we study the speed of information spreading in dynamic networks. Our aim is to introduce a general framework where simple communication primitives can be analysed, to achieve a better understanding of the main features of dynamic networks. The thesis provides a revised version of the results presented in [6, 5, 7]. We here give an informal overall description of such results. One of the simplest communication tasks that seems worth studying in dynamic networks is the broadcast problem: one distinguished node of the network (the source node) aims to send a message to all the other nodes of the network. We model a dynamic network as an evolving graph. An evolving graph is a sequence of graphs (i.e. the snapshots) with the same set of nodes. To analyse the speed of information spreading in evolving graphs, we use a communication process that is usually called flooding (or flooding mechanism). The flooding process can be described as follows: start with one single informed node (the source), and assume that when a non-informed node v has an informed neighbour then node v becomes informed itself, at the next time step. The question is how long does it take to get all the nodes informed? Given an evolving graph and a source node, the completion time of the flooding process is the first time step in which all the nodes are informed. The flooding time of an evolving graph is the maximum completion time over all possible sources. The flooding time of a static graph equals the diameter of the graph; in particular it is finite if and only if the graph is connected. Can we say something like that for evolving graphs? Is it true that, for example, if every snapshot of an evolving graph G has small diameter then the flooding time of G is small? In order to investigate such issue in a clean way, let us introduce the concept of adversarial evolving graph.