On the cover time and mixing time of random geometric graphs

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 h...

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Bibliographic Details
Published in:Theoretical computer science 2007-06, Vol.380 (1), p.2-22
Main Authors: Avin, Chen, Ercal, Gunes
Format: Article
Language:English
Subjects:
Applied sciences
Combinatorics
Combinatorics. Ordered structures
Computer science
control theory
systems
Cover time
Exact sciences and technology
Global analysis, analysis on manifolds
Graph theory
Information retrieval. Graph
Mathematics
Miscellaneous
Mixing time
Random graphs
Random walks
Sciences and techniques of general use
Theoretical computing
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
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Summary: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.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2007.02.065