61 research outputs found

### 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

### First Passage Percolation on the Erdős–Rényi Random Graph

In this paper we explore first passage percolation (FPP) on the Erdos-Renyi 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 for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to λ/(λ − 1) log n. Furthermore, we prove that the minimal weight centered by log n/(λ − 1) converges in distribution. We also investigate the dense regime, where npn → ∞. We find that although the base graph is a ultra small (meaning that graph distances between uniformly chosen vertices are o(log n)), attaching random edge weights changes the geometry of the network completely. Indeed, the hopcount Hn satisfies the universality property that whatever be the value of pn, Hn/ log n → 1 in probability and, more precisely, (Hn−β n log n)/√log n, where β n = λn/(λn− 1), has a limiting standard normal distribution. The constant β n can be replaced by 1 precisely when λn ≫ √log n, a case that has appeared in the literature (under stronger conditions on λn) in [2, 12]. We also find bounds for the maximal weight and maximal hopcount between vertices in the graph. This paper continues the investigation of FPP initiated in [2] and [3]. Compared to the setting on the configuration model studied in [3], the proofs presented here are much simpler due to a direct relation between FPP on the Erdos-Renyi random graph and thinned continuous-time branching processes

### First passage percolation on random graphs with finite mean degrees

We study first passage percolation on the configuration model. Assuming that
each edge has an independent exponentially distributed edge weight, we derive
explicit distributional asymptotics for the minimum weight between two randomly
chosen connected vertices in the network, as well as for the number of edges on
the least weight path, the so-called hopcount. We analyze the configuration
model with degree power-law exponent $\tau>2$, in which the degrees are assumed
to be i.i.d. with a tail distribution which is either of power-law form with
exponent $\tau-1>1$, or has even thinner tails ($\tau=\infty$). In this model,
the degrees have a finite first moment, while the variance is finite for
$\tau>3$, but infinite for $\tau\in(2,3)$. We prove a central limit theorem for
the hopcount, with asymptotically equal means and variances equal to
$\alpha\log{n}$, where $\alpha\in(0,1)$ for $\tau\in(2,3)$, while $\alpha>1$
for $\tau>3$. Here $n$ denotes the size of the graph. For $\tau\in (2,3)$, it
is known that the graph distance between two randomly chosen connected vertices
is proportional to $\log \log{n}$ [Electron. J. Probab. 12 (2007) 703--766],
that is, distances are ultra small. Thus, the addition of edge weights causes a
marked change in the geometry of the network. We further study the weight of
the least weight path and prove convergence in distribution of an appropriately
centered version. This study continues the program initiated in [J. Math. Phys.
49 (2008) 125218] of showing that $\log{n}$ is the correct scaling for the
hopcount under i.i.d. edge disorder, even if the graph distance between two
randomly chosen vertices is of much smaller order. The case of infinite mean
degrees ($\tau\in[1,2)$) is studied in [Extreme value theory,
Poisson--Dirichlet distributions and first passage percolation on random
networks (2009) Preprint] where it is proved that the hopcount remains
uniformly bounded and converges in distribution.Comment: Published in at http://dx.doi.org/10.1214/09-AAP666 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org

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