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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: [email protected]
∗∗ Postal address: Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
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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
Type
Research Article
Information
Advances in Applied Probability , Volume 47 , Issue 2 , June 2015 , pp. 589 - 610
Copyright
© Applied Probability Trust 

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