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<lb>We study first passage percolation on the configuration model. Assuming that each edge has an<lb>independent exponentially distributed edge weight, we derive explicit distributional asymptotics for<lb>the minimum weight between two randomly chosen connected vertices in the network, as well as for<lb>the number of edges on the least weight path, the… (More)

- Remco van der Hofstad, Gerard Hooghiemstra, Piet Van Mieghem
- Random Struct. Algorithms
- 2005

<lb>In this paper we study a random graph with N nodes, where node j has degree Dj and<lb>{Dj}j=1 are i.i.d. with P(Dj ≤ x) = F (x). We assume that 1 − F (x) ≤ cx for some τ > 3<lb>and some constant c > 0. This graph model is a variant of the so-called configuration model,<lb>and includes heavy tail degrees with finite variance.<lb>The minimal number of… (More)

We study first passage percolation on the random graph Gp(N) with exponentially distributed weights on the links. For the special case of the complete graph this problem can be described in terms of a continuous time Markov chain and recursive trees. The Markov chain X(t) describes the number of nodes that can be reached from the initial node in time t. The… (More)

- Shankar Bhamidi, Remco van der Hofstad, Gerard Hooghiemstra
- Combinatorics, Probability & Computing
- 2011

In this paper we explore first passage percolation (FPP) on the Erdős-Rényi random graph Gn(pn), where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when npn → λ > 1, we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as… (More)

- Piet Van Mieghem, Gerard Hooghiemstra, Remco van der Hofstad
- IEEE/ACM Trans. Netw.
- 2001

The average number of joint hops in a shortest-path multicast tree from a root to <i>m</i> arbitrary chosen group member nodes is studied. A general theory for all graphs, hence including the graph representation of the Internet, is presented which quantifies the multicast reduction in network links compared to <i>m</i> times unicast. For two special types… (More)

- Remco van der Hofstad, Gerard Hooghiemstra, Piet Van Mieghem
- Random Struct. Algorithms
- 2002

In this paper we study the covariance structure of the number of nodes k and l steps away from the root in random recursive trees. We give an analytic expression valid for all k, l and tree sizes N . The fraction of nodes k steps away from the root is a random probability distribution in k. The expression for the covariances allows us to show that the total… (More)

We consider first passage percolation on sparse random graphs with prescribed degree distributions and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform X logX-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical… (More)

In this paper we study typical distances in random graphs with i.i.d. degrees of which the tail of the common distribution function is regularly varying with exponent 1− τ . Depending on the value of the parameter τ we can distinct three cases: (i) τ > 3, where the degrees have finite variance, (ii) τ ∈ (2, 3), where the degrees have infinite variance, but… (More)

- Henri van den Esker, Remco van der Hofstad, Gerard Hooghiemstra, Dmitri Znamenski
- 2004

We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function F is regularly varying with exponent τ ∈ [1, 2]. In particular, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal number of edges between two arbitrary nodes, also… (More)

In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving… (More)