Random Walks on Random Graphs

The aim of this article is to discuss some of the notions and applications of random walks on finite graphs, especially as they apply to random graphs. In this section we give some basic definitions, in Section 2 we review applications of random walks in computer science, and in Section 3 we focus on walks in random graphs.

[1]  G. Pólya.  Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz , 1921 .

[2]  Richard J. Lipton,et al.  Random walks, universal traversal sequences, and the complexity of maze problems , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

[3]  Peter G. Doyle,et al.  Random Walks and Electric Networks: REFERENCES , 1987 .

[4]  N. Alon.  Eigenvalues and expanders , 1986, Comb..

[5]  János Komlós,et al.  Deterministic simulation in LOGSPACE , 1987, STOC.

[6]  Eli Upfal,et al.  Trading space for time in undirected s-t connectivity , 1989, STOC '89.

[7]  David J. Aldous,et al.  Lower bounds for covering times for reversible Markov chains and random walks on graphs , 1989 .

[8]  Andrei Z. Broder,et al.  Generating random spanning trees , 1989, 30th Annual Symposium on Foundations of Computer Science.

[9]  David Aldous,et al.  The Random Walk Construction of Uniform Spanning Trees and Uniform Labelled Trees , 1990, SIAM J. Discret. Math..

[10]  Martin E. Dyer,et al.  Random walks, totally unimodular matrices, and a randomised dual simplex algorithm , 1994, IPCO.

[11]  Uriel Feige,et al.  A Tight Upper Bound on the Cover Time for Random Walks on Graphs , 1995, Random Struct. Algorithms.

[12]  Mark Jerrum,et al.  A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph , 1995, Random Struct. Algorithms.

[13]  Uriel Feige,et al.  A Tight Lower Bound on the Cover Time for Random Walks on Graphs , 1995, Random Struct. Algorithms.

[14]  Mark Jerrum,et al.  The Markov chain Monte Carlo method: an approach to approximate counting and integration , 1996 .

[15]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[16]  Johan Jonasson.  On the Cover Time for Random Walks on Random Graphs , 1998, Comb. Probab. Comput..

[17]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[18]  Alan M. Frieze.  Edge-disjoint paths in expander graphs , 2000, SODA '00.

[19]  Alan M. Frieze,et al.  Crawling on Simple Models of Web Graphs , 2004, Internet Math..

[20]  Alan M. Frieze,et al.  The cover time of sparse random graphs. , 2003, SODA '03.

[21]  Chen Avin,et al.  On the Cover Time of Random Geometric Graphs , 2005, ICALP.

[22]  Omer Reingold,et al.  Undirected ST-connectivity in log-space , 2005, STOC '05.

[23]  Alan M. Frieze,et al.  The Cover Time of Random Regular Graphs , 2005, SIAM J. Discret. Math..

[24]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[25]  Alan M. Frieze,et al.  The cover time of the preferential attachment graph , 2007, J. Comb. Theory, Ser. B.

[26]  Alan M. Frieze,et al.  The Cover Time of Random Digraphs , 2007, APPROX-RANDOM.

[27]  Chen Avin,et al.  On the cover time and mixing time of random geometric graphs , 2007, Theor. Comput. Sci..

[28]  Colin Cooper,et al.  A randomized algorithm for the joining protocol in dynamic distributed networks , 2008, Theor. Comput. Sci..

[29]  Alan M. Frieze,et al.  The cover time of the giant component of a random graph , 2008, Random Struct. Algorithms.

[30]  Alan M. Frieze,et al.  Multiple Random Walks in Random Regular Graphs , 2009, SIAM J. Discret. Math..