On the Cover Time of Random Geometric Graphs [chapter]

Chen Avin, Gunes Ercal
2005 Lecture Notes in Computer Science  
The 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 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
more » ... se 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 ≥ r opt G(n, r) has optimal cover time Θ(n log n) with high probability, and, importantly, the critical radius guaranteeing optimal cover time is on the same order as the critical radius guaranteeing connectivity, namely 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 G(n, r) via computing the power of certain constructed flows. 1 We focus on the 2-dimensional case although we can also generalize our results to any d dimensions.
doi:10.1007/11523468_55 fatcat:poeylyoy7fdzpojq33yyyf4h4i