Structural and Algorithmic Properties of Static and Mobile Random Geometric Graphs

We study fundamental problems for static and mobile networks./nFirst, we consider the random geometric graph model, which is a well-known model for static/nwireless networks. In this model,/nn nodes are distributed independently and uniformly at random/nin the d-dimensional torus of volume n and edges are added between/npairs of nodes whose Euclidean distance is at most some parameter r. We consider the case where r is a sufficiently large constant/nso that a so-called giant component (a connected component with \Theta(n) nodes) exists with high probability./nIn this setting, we show that the graph distance between every pair of/nnodes whose Euclidean distance is sufficiently large is only a constant factor larger than their Euclidean distance./nThis result gives, as a corollary, that the diameter of the giant component is \Theta(n^{1/d}/r)./nThen, we apply this result to analyze the performance of a broadcast algorithm known as the push algorithm./nIn this algorithm, at each discrete time step, each informed node chooses/na neighbor independently and uniformly at random and informs it. We show that the push algorithm informs all nodes of the giant component/nof a random geometric graph within a number of steps that is only a constant factor larger then the diameter of the giant component./nIn the second part of the thesis, we consider a model of mobile graphs that we call mobile geometric graphs, and which is an extension of/nthe random geometric graph model to the setting where nodes are not static but are moving in space in continuous time./nIn this model, we start with a random geometric graph and let the nodes move as independent Brownian motions./nThen, at any given time, there exists an edge between every pair of nodes whose Euclidean distance at that time/nis at most r. This model has been recently used as a model for mobile wireless networks./nWe study four fundamental problems in this model: detection (the time until a target point---fixed or moving---is/nwithin distance r of some node of the graph); coverage (the time until all points inside/na finite box are detected by the graph);/npercolation (the time until a given node belongs to the giant component of the graph) and/nbroadcast (the time until all nodes of the graph receive a piece of information that was initially known by a single node)./nWe obtain precise asymptotics for these quantities by combining ideas from stochastic geometry,/ncoupling and multi-scale analysis./nFinally, in the last part of the thesis, we revisit the push algorithm described above and study its performance in general regular graphs./nOur goal is to understand the relation between the performance of the push algorithm and the vertex expansion of the graph./nWe prove an upper bound for the runtime of this algorithm that depends on the vertex expansion of the graph and is tight up to polylogarithmic factors./nThen, we show that there exists a substantial difference between the impact of vertex expansion and edge expansion on the performance of the push algorithm./nIn particular, we prove the existence of regular graphs (which are also vertex transitive) that have constant vertex expansion and for which the runtime of/nthe push algorithm is a factor of \Omega(log n) slower than on any regular graph with constant edge expansion.

Advisor: Sinclair, Alistair