Extreme value theory, Poisson-Dirichlet distributions and FPP on random networks

We study first passage percolation on the configuration model (CM) having power-law degrees with exponent $\tau\in [1,2)$. To this end we equip the edges with exponential 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, which can be computed in terms of the Poisson-Dirichlet distribution. We explicitly describe these limits via the construction of an infinite limiting object describing the FPP problem in the densely connected core of the network. We consider two separate cases, namely, the {\it original CM}, in which each edge, regardless of its multiplicity, receives an independent exponential weight, as well as the {\it 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 $\tau>2$, 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 one can transfer information 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

Distances in random graphs with finite mean and infinite variance degrees

In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent $\tau\in (2,3)$. The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph with $N$ nodes is investigated when $N\to \infty$. When $\tau\in (2,3)$, this graph distance grows like $2\frac{\log\log N}{|\log(\tau-2)|}$. In different papers, the cases $\tau>3$ and $\tau\in (1,2)$ have been studied. We also study the fluctuations around these asymptotic means, and describe their distributions. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.Comment: 52 pages, 4 figure

Distances in random graphs with infinite mean degrees

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 $\tau\in (1,2)$. Thus, 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 called the graph distance or the hopcount, in a graph with $N$ nodes is investigated when $N\to \infty$. The paper is part of a sequel of three papers. The other two papers study the case where $\tau \in (2,3)$, and $\tau \in (3,\infty),$ respectively. The main result of this paper is that the graph distance converges for $\tau\in (1,2)$ to a limit random variable with probability mass exclusively on the points 2 and 3. We also consider the case where we condition the degrees to be at most $N^{\alpha}$ for some $\alpha>0.$ For $\tau^{-1}<\alpha<(\tau-1)^{-1}$, the hopcount converges to 3 in probability, while for $\alpha>(\tau-1)^{-1}$, the hopcount converges to the same limit as for the unconditioned degrees. Our results give convincing asymptotics for the hopcount when the mean degree is infinite, using extreme value theory.Comment: 20 pages, 2 figure

Weak disorder in the stochastic mean-field model of distance II

In this paper, we study the complete graph $K_n$ with n vertices, where we attach an independent and identically distributed (i.i.d.) weight to each of the n(n-1)/2 edges. We focus on the weight $W_n$ and the number of edges $H_n$ of the minimal weight path between vertex 1 and vertex n. It is shown in (Ann. Appl. Probab. 22 (2012) 29-69) that when the weights on the edges are i.i.d. with distribution equal to that of $E^s$, where $s>0$ is some parameter, and E has an exponential distribution with mean 1, then $H_n$ is asymptotically normal with asymptotic mean $s\log n$ and asymptotic variance $s^2\log n$. In this paper, we analyze the situation when the weights have distribution $E^{-s},s>0$, in which case the behavior of $H_n$ is markedly different as $H_n$ is a tight sequence of random variables. More precisely, we use the method of Stein-Chen for Poisson approximations to show that, for almost all $s>0$, the hopcount $H_n$ converges in probability to the nearest integer of s+1 greater than or equal to 2, and identify the limiting distribution of the recentered and rescaled minimal weight. For a countable set of special s values denoted by $\mathcal{S}=\{s_j\}_{j\geq2}$, the hopcount $H_n$ takes on the values j and j+1 each with positive probability.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ402 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm