TY - JOUR

T1 - On the cover time and mixing time of random geometric graphs

AU - Avin, Chen

AU - Ercal, Gunes

N1 - Funding Information:
The authors would like to thank Shailesh Vaya, Eli Gafni, and Adam Meyerson for helpful discussions and David Dayan-Rosenman and the anonymous reviewers for their comments and corrections. The first author acknowledges partial support from ONR (MURI) grant #N00014-00-1-0617 and from the Department of Communication System Engineering at Ben-Gurion University, Israel.

PY - 2007/6/21

Y1 - 2007/6/21

N2 - 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 ropt such that for any r ≥ roptG (n, r) has optimal cover time of Θ (n log n) with high probability, and, importantly, 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). On the other hand, the radius required for rapid mixing rrapid = ω (rcon), and, in particular, rrapid = 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.

AB - 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 ropt such that for any r ≥ roptG (n, r) has optimal cover time of Θ (n log n) with high probability, and, importantly, 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). On the other hand, the radius required for rapid mixing rrapid = ω (rcon), and, in particular, rrapid = 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.

KW - Cover time

KW - Mixing time

KW - Random graphs

KW - Random walks

UR - http://www.scopus.com/inward/record.url?scp=34248512089&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2007.02.065

DO - 10.1016/j.tcs.2007.02.065

M3 - Article

AN - SCOPUS:34248512089

SN - 0304-3975

VL - 380

SP - 2

EP - 22

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 1-2

ER -