On the cover time and mixing time of random geometric graphs

Abstract

The cover time and mixing 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 r opt such that for any r ≥ r opt 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 ) . On the other hand, the radius required for rapid mixing r rapid = ω ( r con ) , and, in particular, r rapid = Θ ( 1 / poly ( log n ) ) . We are able to draw our results by giving a tight bound on the electrical resistance and conductance of G ( n , r ) via certain constructed flows.