The cover time of the preferential attachment graph

The preferential attachment graph G"m(n) is a random graph formed by adding a new vertex at each time step, with m edges which point to vertices selected at random with probability proportional to their degree. Thus at time n there are n vertices and mn edges. This process yields a graph which has been proposed as a simple model of the world wide web [A. Barabasi, R. Albert, Emergence of scaling in random networks, Science 286 (1999) 509-512]. In this paper we show that if m>=2 then whp the cover time of a simple random walk on G"m(n) is asymptotic to 2mm-1nlogn.

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

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

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

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

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

[6]  W. Hoeffding.  Probability Inequalities for sums of Bounded Random Variables , 1963 .

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

[8]  H. Wilf.  generatingfunctionology: Third Edition , 1990 .

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

[10]  Herbert S. Wilf,et al.  Generating functionology , 1990 .

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

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

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

[14]  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).

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

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