Localization transition of biased random walks on random networks.

We study random walks on large random graphs that are biased towards a randomly chosen but fixed target node. We show that a critical bias strength bc exists such that most walks find the target within a finite time when b > bc. For b < bc, a finite fraction of walks drift off to infinity before hitting the target. The phase transition at b=bc is a critical point in the sense that quantities such as the return probability P(t) show power laws, but finite-size behavior is complex and does not obey the usual finite-size scaling ansatz. By extending rigorous results for biased walks on Galton-Watson trees, we give the exact analytical value for bc and verify it by large scale simulations.

[1]  R. Lyons.  Random Walks and Percolation on Trees , 1990 .

[2]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[3]  Cohen,et al.  Resilience of the internet to random breakdowns , 2000, Physical review letters.

[4]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

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

[6]  Jean-Pierre Bourguignon,et al.  Mathematische Annalen , 1893 .

[7]  Walter Willinger,et al.  Scaling phenomena in the Internet: Critically examining criticality , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[8]  M. Rosvall,et al.  Self-assembly of information in networks , 2006 .

[9]  D. Dhar.  Diffusion and drift on percolation networks in an external field , 1984 .

[10]  Bruce A. Reed,et al.  A Critical Point for Random Graphs with a Given Degree Sequence , 1995, Random Struct. Algorithms.

[11]  Russell Lyons,et al.  Biased random walks on Galton–Watson trees , 1996 .

[12]  S. Havlin,et al.  Scale-free networks are ultrasmall. , 2002, Physical review letters.

[13]  F. Chung,et al.  The average distances in random graphs with given expected degrees , 2002, Proceedings of the National Academy of Sciences of the United States of America.

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

[15]  Transition to localization of biased walkers in a randomly absorbing environment , 2001, cond-mat/0107525.