On the Cover Time of Random Geometric Graphs

Abstract

AbstractThe cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of ad-hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph \({\mathcal G}(n,r)\) is obtained by placing n points uniformly at random on the unit square and connecting two points iff their Euclidean distance is at most r. The phase transition behavior with respect to the radius r of such graphs has been of special interest. We show that there exists a critical radius r opt such that for any \(r \geq r_{\rm opt} {\mathcal G}(n,r)\) has optimal cover time of Θ(n log n) with high probability, and, importantly, r opt = Θ(r con) where r con denotes the critical radius guaranteeing asymptotic connectivity. Moreover, since a disconnected graph has infinite cover time, there is a phase transition and the corresponding threshold width is O(r con). We are able to draw our results by giving a tight bound on the electrical resistance of \({\mathcal G}(n,r)\) via the power of certain constructed flows.KeywordsSensor NetworkRandom WalkWireless Sensor NetworkRandom GraphCritical RadiusThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.