On the Cover Time of Random Geometric Graphs

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 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 ropt such that for any r ≥ ropt 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 ropt = Θ(rcon) where rcon 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(rcon). 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.