Shortest-Weight Paths in Random Regular Graphs

Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an independent and identically distributed exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d \geq 3$, we show that the longest of these shortest-weight paths has about $\widehat{\alpha} \log n$ edges, where $\widehat{\alpha} $ is the unique solution of the equation $\alpha \log\big(\frac{d-2}{d-1}\alpha\big) - \alpha = \frac{d-3}{d-2}$ for $\alpha > \frac{d-1}{d-2}$.

[1]  Remco van der Hofstad,et al.  First Passage Percolation on the Erdős–Rényi Random Graph , 2010, Combinatorics, Probability and Computing.

[2]  Robin Pemantle,et al.  First passage percolation and a model for competing spatial growth , 1997 .

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

[4]  Luc Devroye,et al.  The height of increasing trees , 2008, Random Struct. Algorithms.

[5]  Justin Salez,et al.  Joint distribution of distances in random regular networks , 2012 .

[6]  Yuval Peres,et al.  Diameters in Supercritical Random Graphs Via First Passage Percolation , 2009, Combinatorics, Probability and Computing.

[7]  Gábor Lugosi,et al.  The Longest Minimum-Weight Path in a Complete Graph , 2010, Comb. Probab. Comput..

[8]  Alan M. Frieze,et al.  The shortest-path problem for graphs with random arc-lengths , 1985, Discret. Appl. Math..

[9]  Shankar Bhamidi,et al.  Edge flows in the complete random-lengths network , 2010, Random Struct. Algorithms.

[10]  Svante Janson,et al.  One, Two and Three Times log n/n for Paths in a Complete Graph with Random Weights , 1999, Combinatorics, Probability and Computing.

[11]  H. Kesten.  Aspects of first passage percolation , 1986 .

[12]  Jian Ding,et al.  Scaling window for mean-field percolation of averages , 2011, 1110.3361.

[13]  Harry Kesten,et al.  First-passage percolation, network flows and electrical resistances , 1984 .

[14]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[15]  Marc Lelarge,et al.  Flooding in Weighted Random Graphs , 2010, ANALCO 2011.

[16]  N. Wormald,et al.  Models of the , 2010 .

[17]  Allan Sly,et al.  Cutoff phenomena for random walks on random regular graphs , 2008, 0812.0060.

[18]  Luc Devroye,et al.  The height of increasing trees , 2008 .

[19]  G. Hooghiemstra,et al.  First passage percolation on random graphs with finite mean degrees , 2009, 0903.5136.

[20]  Yuval Peres,et al.  All-Pairs Shortest Paths in O(n2) Time with High Probability , 2010, FOCS.

[21]  N. Wormald.  Models of random regular graphs , 2010 .

[22]  Edward A. Bender,et al.  The Asymptotic Number of Labeled Graphs with Given Degree Sequences , 1978, J. Comb. Theory A.

[23]  Marc Lelarge,et al.  The diameter of weighted random graphs , 2011, 1112.6330.

[24]  Shankar Bhamidi,et al.  First passage percolation on locally treelike networks. I. Dense random graphs , 2008 .