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Joint Distribution of Distances in Large Random Regular Networks

Part of: Combinatorial probability Operations research and management science Graph theory

Published online by Cambridge University Press:  30 January 2018

Justin Salez
Justin Salez*
Affiliation:
University of California, Berkeley
*
∗ Current address: Université Paris Diderot, LPMA, Site Chevaleret, case 7012, 75205 Paris Cedex 13, France. Email address: justin.salez@univ-paris-diderot.fr
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Abstract

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We study the array of point-to-point distances in random regular graphs equipped with exponential edge lengths. We consider the regime where the degree is kept fixed while the number of vertices tends to ∞. The marginal distribution of an individual entry is now well understood, thanks to the work of Bhamidi, van der Hofstad and Hooghiemstra (2010). The purpose of this note is to show that the whole array, suitably recentered, converges in the weak sense to an explicit infinite random array. Our proof consists in analyzing the invasion of the network by several mutually exclusive flows emanating from different sources and propagating simultaneously along the edges.

Keywords

Random regular graphdistance matrixfirst passage percolationmultitype Richardson processconfiguration modelbranching process approximation

MSC classification

Primary: 60C05: Combinatorial probability
Secondary: 05C80: Random graphs 90B15: Network models, stochastic
Type
Research Article
Information
Journal of Applied Probability , Volume 50 , Issue 3 , September 2013 , pp. 861 - 870
Copyright
© Applied Probability Trust 

References

Aldous, D. J. (2010). More uses of exchangeability: representations of complex random structures. In Probability and Mathematical Genetics (London Math. Soc. Lecture Notes 378), Cambridge University Press, pp. 35–63.CrossRefGoogle Scholar
Aldous, D. J. and Bhamidi, S. (2010). Edge flows in the complete random-lengths network. Random Structures Algorithms 37, 271–311.CrossRefGoogle Scholar
Amini, A. and Lelarge, M. (2011). The diameter of weighted random graphs. Preprint. Available at http://arXiv.org/abs/1112.6330v1.Google Scholar
Antunović, T., Dekel, Y., Mossel, E. and Peres, Y. (2011). Competing first passage percolation on random regular graphs. Preprint. Available at http://arXiv.org/abs/1109.2575v1.Google Scholar
Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover Publications, Mineola, NY.Google Scholar
Bhamidi, S. (2008). First passage percolation on locally treelike networks. I. Dense random graphs. J. Math. Phys. 49, 125218, 27 pp.CrossRefGoogle Scholar
Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). Extreme value theory, Poisson-Dirichlet distributions, and first passage percolation on random networks. Adv. Appl. Prob. 42, 706–738.CrossRefGoogle Scholar
Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Prob. 20, 1907–1907.CrossRefGoogle Scholar
Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2011). First passage percolation on the Erdős-{R}ényi random graph. Combinatorics Prob. Comput. 20, 683–707.CrossRefGoogle Scholar
Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2012). Universality for first passage percolation on sparse random graphs. Preprint. Available at http://arXiv.org/abs/1210.6839v1.Google Scholar
Bollobás, B. (1980). A probabilistic proof of an asymptotic formula for the number of labelled regular graphs. Europ. J. Combinatorics 1, 311–316.CrossRefGoogle Scholar
Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Prob. 9, 583–603.CrossRefGoogle Scholar
Deijfen, M. and Häggström, O. (2006). The initial configuration is irrelevant for the possibility of mutual unbounded growth in the two-type Richardson model. Combinatorics Prob. Comput. 15, 345–353.CrossRefGoogle Scholar
Deijfen, M. and Häggström, O. (2006). Nonmonotonic coexistence regions for the two-type Richardson model on graphs. Electron. J. Prob. 11, 331–344.CrossRefGoogle Scholar
Deijfen, M. and Häggström, O. (2007). The two-type Richardson model with unbounded initial configurations. Ann. Appl. Prob. 17, 1639–1656.CrossRefGoogle Scholar
Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. (2010). Diameters in supercritical random graphs via first passage percolation. Combinatorics Prob. Comput. 19, 729–751.CrossRefGoogle Scholar
Garet, O. and Marchand, R. (2008). First-passage competition with different speeds: positive density for both species is impossible. Electron. J. Prob. 13, 2118–2159.CrossRefGoogle Scholar
Häggström, O. and Pemantle, R. (1998). First passage percolation and a model for competing spatial growth. J. Appl. Prob. 35, 683–692.CrossRefGoogle Scholar
Häggström, O. and Pemantle, R. (2000). Absence of mutual unbounded growth for almost all parameter values in the two-type Richardson model. Stochastic Process. Appl. 90, 207–222.CrossRefGoogle Scholar
Hoffman, C. (2005). Coexistence for Richardson type competing spatial growth models. Ann. Appl. Prob. 15, 739–747.CrossRefGoogle Scholar
Hoffman, C. (2008). Geodesics in first passage percolation. Ann. Appl. Prob. 18, 1944–1944.CrossRefGoogle Scholar
Janson, S. (1999). One, two and three times log n/n for paths in a complete graph with random weights. Combinatorics Prob. Comput. 8, 347–361.CrossRefGoogle Scholar
Janson, S. (2009). The probability that a random multigraph is simple. Combinatorics Prob. Comput. 18, 205–225.CrossRefGoogle Scholar
Joseph, A. (2011). The component sizes of a critical random graph with given degree sequence. Preprint. Available at http://arXiv.org/abs/1012.2352v2.Google Scholar
Kesten, H. (1986). Aspects of first passage percolation. In École d'été de Probabilités de Saint-Flour, XIV—1984 (Lecture Notes Math. 1180), Springer, Berlin, pp. 125–264.CrossRefGoogle Scholar
Van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2001). First-passage percolation on the random graph. Prob. Eng. Inf. Sci. 15, 225–237.CrossRefGoogle Scholar
Van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2005). Distances in random graphs with finite variance degrees. Random Structures Algorithms 27, 76–123.CrossRefGoogle Scholar