The cover time of two classes of random graphs

Let <i>G</i> = (<i>V,E</i>) be a connected graph, let |<i>V</i>| = <i>n</i>, and |<i>E</i>| = <i>m.</i> <i>A random walk W<inf>u</inf>, u</i> ∈ <i>V</i> on the undirected graph <i>G</i> = (<i>V, E</i>) is a Markov chain <i>X<inf>0</inf></i> = <i>u, X<inf>1</inf>,...X<inf>t</inf>,...</i> ∈ <i>V</i> associated to a particle that moves from vertex to vertex according to the following rule: the probability of a transition from vertex <i>i</i>, of degree <i>d<inf>i</inf></i>, to vertex <i>j</i> is 1/<i>d<inf>i</inf></i> if {<i>i,j</i>} ∈ <i>E</i>, and 0 otherwise. For <i>u</i> ∈ <i>V</i> let <i>C<inf>u</inf></i> be the expected time taken for <i>W<inf>u</inf></i> to visit every vertex of <i>G.</i> The <i>cover time C<inf>G</inf></i> of <i>G</i> is defined as <i>C<inf>G</inf></i> = max<i>u</i>∈<i>V C<inf>u</inf>.</i> The cover time of connected graphs has been extensively studied. It is a classic result of Aleliunas, Karp, Lipton, Lovász and Rackoff [2] that <i>C<inf>G</inf></i> ≤ 2<i>m</i>(<i>n</i> - 1). It was shown by Feige [11], [12], that for any connected graph <i>G</i>[EQUATION]The lower bound is achieved by (for example) the complete graph <i>K<inf>n</inf></i>, whose cover time is determined by the Coupon Collector problem.

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

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

[3]  D. Aldous.  On the time taken by random walks on finite groups to visit every state , 1983 .

[4]  Béla Bollobás,et al.  Random Graphs , 1985 .

[5]  Béla Bollobás,et al.  The degree sequence of a scale‐free random graph process , 2001, Random Struct. Algorithms.

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

[7]  P. A. P. Moran,et al.  An introduction to probability theory , 1968 .

[8]  N. Alon.  Tools from higher algebra , 1996 .

[9]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[10]  Béla Bollobás,et al.  A Probabilistic Proof of an Asymptotic Formula for the Number of Labelled Regular Graphs , 1980, Eur. J. Comb..

[11]  J. W. Brown,et al.  Complex Variables and Applications , 1985 .

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

[13]  Amin Saberi,et al.  On certain connectivity properties of the Internet topology , 2003, 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings..

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

[15]  Béla Bollobás,et al.  The Diameter of a Scale-Free Random Graph , 2004, Comb..

[16]  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).

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

[18]  L. Lovász.  Combinatorial problems and exercises , 1979 .