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First Passage Percolation on Inhomogeneous Random Graphs

Part of: Operations research and management science Markov processes Graph theory

Published online by Cambridge University Press:  22 February 2016

István Kolossváry  and
Júlia Komjáthy
István Kolossváry*
Affiliation:
Budapest University of Technology and Economics
Júlia Komjáthy*
Affiliation:
Budapest University of Technology and Economics and Eindhoven University of Technology
*
∗ Postal address: Budapest University of Technology and Economics, Inter-University Centre for Telecommunications and Informatics, 4028 Debrecen, Kassai út 26, Hungary. Email address: istvanko@math.bme.hu
∗∗ Postal address: Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Email address: j.komjathy@tue.nl
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Abstract

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In this paper we investigate first passage percolation on an inhomogeneous random graph model introduced by Bollobás et al. (2007). Each vertex in the graph has a type from a type space, and edge probabilities are independent, but depend on the types of the end vertices. Each edge is given an independent exponential weight. We determine the distribution of the weight of the shortest path between uniformly chosen vertices in the giant component and show that the hopcount, i.e. the number of edges on this minimal-weight path, properly normalized, follows a central limit theorem. We handle the cases where the average number of neighbors λ̃n of a vertex tends to a finite λ̃ in full generality and consider λ̃ = ∞ under mild assumptions. This paper is a generalization of the paper of Bhamidi et al. (2011), where first passage percolation is explored on the Erdős-Rényi graphs.

Keywords

Inhomogeneous random graphshortest-weight pathhopcountfirst passage percolationcontinuous-time multitype branching process

MSC classification

Primary: 05C80: Random graphs
Secondary: 90B15: Network models, stochastic 60J85: Applications of branching processes

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 589 - 610
Copyright
© Applied Probability Trust 

References

Amini, H. and Lelarge, M. (2015). The diameter of weighted random graphs. Ann. Appl. Prob. 3, 1686–1727.Google Scholar
Asmussen, S. (1977). Almost sure behavior of linear functionals of supercritical branching processes. Trans. Amer. Math. Soc. 231, 233–248.Google Scholar
Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.Google Scholar
Barbour, A. D. and Reinert, G. (2013). Approximating the epidemic curve. Electron. J. Prob. 18, 30pp.Google Scholar
Bhamidi, S. (2008). First passage percolation on locally treelike networks. I. Dense random graphs. J. Math. Phys. 49, 125–218.Google Scholar
Bhamidi, S. and van der Hofstad, R. (2012). Weak disorder asymptotics in the stochastic mean-field model of distance. Ann. Appl. Prob. 22, 29–69.Google 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.Google Scholar
Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2011). First passage percolation on the Erdős–Rényi random graph. Combin. Prob. Comput. 20, 683–707.Google 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/1210.6839.Google Scholar
Bhamidi, S., van der Hofstad, R. and Komjáthy, J. (2014). The front of the epidemic spread and first passage percolation. J. Appl. Prob. 51, 101–121.Google Scholar
Bollobás, B. and Fernandez de la Vega, W. (1982). The diameter of random regular graphs. Combinatorica 2, 125–134.Google Scholar
Bollobás, B., Janson, S. and Riordan, O. (2007). The phase transition in inhomogeneous random graphs. Random Structures Algorithms 31, 3–122.Google Scholar
Britton, T., Deijfen, M. and Martin-Löf, A. (2006). Generating simple random graphs with prescribed degree distribution. J. Statist. Phys. 124, 1377–1397.Google Scholar
Bühler, W. J. (1971). Generations and degree of relationship in supercritical Markov branching processes. Prob. Theory Relat. Fields 18, 141–152.Google Scholar
Chung, F. and Lu, L. (2002). Connected components in random graphs with given expected degree sequences. Ann. Comb. 6, 125–145.Google Scholar
Chung, F. and Lu, L. (2003). The average distance in a random graph with given expected degrees. Internet Math. 1, 91–113.Google Scholar
Fernholz, D. and Ramachandran, V. (2007). The diameter of sparse random graphs. Random Structures Algorithms 31, 482–516.Google Scholar
Howard, C. D. (2004). Models of first-passage percolation. In Probability on Discrete Structures, Springer, Berlin, pp. 125–173.CrossRefGoogle Scholar
Janson, S. (1999). One, two and three times log n/n for paths in a complete graph with random weights. Combin. Prob. Comput. 8, 347–361.Google Scholar
Janson, S. (2004). Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Process. Appl. 110, 177–245.Google Scholar
Kharlamov, B. P. (1969). The numbers of generations in a branching process with an arbitrary set of particle types. Theory Prob. Appl. 14, 432–449.Google Scholar
Norros, I. and Reittu, H. (2006). On a conditionally Poissonian graph process. Adv. Appl. Prob. 38, 59–75.Google Scholar
Söderberg, B. (2002). General formalism for inhomogeneous random graphs. Phys. Rev. E 66, 066121.Google 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.Google Scholar
Van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2007). Distances in random graphs with finite mean and infinite variance degrees. Electron. J. Prob. 12, 703–766.Google Scholar