Print Email Facebook Twitter Extreme value theory, Poisson-Dirichlet distributions, and first passage percolation on random networks Title Extreme value theory, Poisson-Dirichlet distributions, and first passage percolation on random networks Author Bhamidi, S. Van der Hofstad, R. Hooghiemstra, G. Faculty Electrical Engineering, Mathematics and Computer Science Department Delft Institute of Applied Mathematics Date 2010-09-01 Abstract We study first passage percolation (FPP) on the configuration model (CM) having power-law degrees with exponent ? ? [1, 2) and exponential edge weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal-weight path, both of which can be computed in terms of the Poisson-Dirichlet distribution. We explicitly describe these limits via construction of infinite limiting objects describing the FPP problem in the densely connected core of the network. We consider two separate cases, the original CM, in which each edge, regardless of its multiplicity, receives an independent exponential weight, and the erased CM, for which there is an independent exponential weight between any pair of direct neighbors. While the results are qualitatively similar, surprisingly, the limiting random variables are quite different. Our results imply that the flow carrying properties of the network are markedly different from either the mean-field setting or the locally tree-like setting, which occurs as ? > 2, and for which the hopcount between typical vertices scales as log n. In our setting the hopcount is tight and has an explicit limiting distribution, showing that information can be transferred remarkably quickly between different vertices in the network. This efficiency has a down side in that such networks are remarkably fragile to directed attacks. These results continue a general program by the authors to obtain a complete picture of how random disorder changes the inherent geometry of various random network models; see Aldous and Bhamidi (2010), Bhamidi (2008), and Bhamidi, van der Hofstad and Hooghiemstra (2009). Subject configuration modelrandom graphfirst passage percolationhopcountextreme value theoryPoisson-Dirichlet distributionscale-free network To reference this document use: http://resolver.tudelft.nl/uuid:951c1c7b-af2c-4b37-9866-1a90758fb8fa DOI https://doi.org/10.1239/aap/1282924060 Publisher Applied Probability Trust ISSN 0001-8678 Source http://projecteuclid.org/euclid.aap/1282924060 Source Advances in Applied Probability, 42 (3), 2010 Part of collection Institutional Repository Document type journal article Rights (c) 2010 Applied Probability Trust Files PDF bhamidi2010.pdf 315.98 KB Close viewer /islandora/object/uuid:951c1c7b-af2c-4b37-9866-1a90758fb8fa/datastream/OBJ/view