Distance graphs: from random geometric graphs to Bernoulli graphs and between
Proceedings of the fifth international workshop on Foundations of mobile computing
Do neighbor-avoiding walkers walk as if in a small-world network?
INFOCOM'09 Proceedings of the 28th IEEE international conference on Computer Communications Workshops
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A random geometric graph G (n, r) is a graph resulting from placing n points uniformly at random on the unit square (or on the unit disk) and connecting two points iff their Euclidean distance is at most the radius r(n). Recently, this class of random graphs has gained relevance as a natural model for wireless ad-hoc and sensor networks. Investigating properties of these graphs can unearth properties of the real-life systems they model and allow for the design of efficient algorithms. In this work, we study properties of random geometric graphs motivated by challenges encounter in sensor networks applications. Sensor networks are constructed from a large number of low-cost, low-power sensors equipped with wireless communication and limited processing capabilities. These devices are expected to be embedded densely into the environment and cooperate to achieve high level tasks. In many cases, the network created by these devices is subject to dramatic structural changes due to failures, node mobility and other factors. Motivated by the disadvantage entailed by topology driven algorithms (e.g. the need to maintain data structures and to execute expensive recovery mechanisms), we investigate algorithms that require no knowledge of network topology, in particular algorithms based on random walks. The investigation we carry out in this dissertation attempts to asses the efficiency of random-walk-based algorithms. Surprisingly we show that despite their simplicity, random-walk-based algorithms can be competitive with optimal topology driven strategies for certain tasks. In particular, we analyze three properties of random walks on these graphs, the mixing time, the cover time and the partial cover time that are essential to determining the efficiency of this approach for sensor network tasks. We also investigated another property of random geometric graphs which has implication for routing and topological control in sensor networks. The goal here is to construct a special subgraph, the Restricted Delaunay Graph, that permits efficient routing, based only on local information. We bound the number of messages needed for this task in these networks and presents a novel algorithm, based on the graph properties, that is more efficient than previous ones. We offer a new extension of random geometric graphs called random distance graphs. to explain some interesting similarities between random geometric graphs and the familiar model of Bernoulli random graphs1. Interestingly while, neither random geometric graphs nor Bernoulli random graphs are suitable to model social networks, a typical case of random distance graphs can captures important properties of social networks. These properties, known as "Small World", includes small average path length and high clustering have been the distinctive mark of many natural networks. 1A Bernoulli random graph (a.k.a. Erdo&huml;s-Rényi graph) B (n, p) is a random graph with n nodes in which each edge is chosen independently at random with an edge probability p(n).