822 research outputs found
Critical behavior in inhomogeneous random graphs
We study the critical behavior of inhomogeneous random graphs where edges are
present independently but with unequal edge occupation probabilities. The edge
probabilities are moderated by vertex weights, and are such that the degree of
vertex i is close in distribution to a Poisson random variable with parameter
w_i, where w_i denotes the weight of vertex i. We choose the weights such that
the weight of a uniformly chosen vertex converges in distribution to a limiting
random variable W, in which case the proportion of vertices with degree k is
close to the probability that a Poisson random variable with random parameter W
takes the value k. We pay special attention to the power-law case, in which
P(W\geq k) is proportional to k^{-(\tau-1)} for some power-law exponent \tau>3,
a property which is then inherited by the asymptotic degree distribution.
We show that the critical behavior depends sensitively on the properties of
the asymptotic degree distribution moderated by the asymptotic weight
distribution W. Indeed, when P(W\geq k) \leq ck^{-(\tau-1)} for all k\geq 1 and
some \tau>4 and c>0, the largest critical connected component in a graph of
size n is of order n^{2/3}, as on the Erd\H{o}s-R\'enyi random graph. When,
instead, P(W\geq k)=ck^{-(\tau-1)}(1+o(1)) for k large and some \tau\in (3,4)
and c>0, the largest critical connected component is of the much smaller order
n^{(\tau-2)/(\tau-1)}.Comment: 26 page
Infinite canonical super-Brownian motion and scaling limits
We construct a measure valued Markov process which we call infinite canonical
super-Brownian motion, and which corresponds to the canonical measure of
super-Brownian motion conditioned on non-extinction. Infinite canonical
super-Brownian motion is a natural candidate for the scaling limit of various
random branching objects on when these objects are (a) critical; (b)
mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the
spread-out oriented percolation incipient infinite cluster above 4 dimensions
and of incipient infinite branching random walk in any dimension. We conjecture
that it also arises as the scaling limit in various other models above the
upper-critical dimension, such as the incipient infinite lattice tree above 8
dimensions, the incipient infinite cluster for unoriented percolation, uniform
spanning trees above 4 dimensions, and invasion percolation above 6 dimensions.
This paper also serves as a survey of recent results linking super-Brownian to
scaling limits in statistical mechanics.Comment: 34 page
Diameter of the stochastic mean-field model of distance
We consider the complete graph \cK_n on vertices with exponential mean
edge lengths. Writing for the weight of the smallest-weight path
between vertex , Janson showed that converges in probability to 3. We extend this result by showing
that converges in distribution to a
limiting random variable that can be identified via a maximization procedure on
a limiting infinite random structure. Interestingly, this limiting random
variable has also appeared as the weak limit of the re-centered graph diameter
of the barely supercritical Erd\H{o}s-R\'enyi random graph in work by Riordan
and Wormald.Comment: 27 page
Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems
We study sample covariance matrices of the form , where
is a matrix with i.i.d. mean zero entries. This is a generalization
of so-called Wishart matrices, where the entries of are independent and
identically distributed standard normal random variables. Such matrices arise
in statistics as sample covariance matrices, and the high-dimensional case,
when is large, arises in the analysis of DNA experiments.
We investigate the large deviation properties of the largest and smallest
eigenvalues of when either is fixed and , or with , in the case where the squares of the
i.i.d. entries have finite exponential moments. Previous results, proving a.s.
limits of the eigenvalues, only require finite fourth moments.
Our most explicit results for large are for the case where the entries of
are with equal probability. We relate the large deviation rate
functions of the smallest and largest eigenvalue to the rate functions for
independent and identically distributed standard normal entries of . This
case is of particular interest, since it is related to the problem of the
decoding of a signal in a code division multiple access system arising in
mobile communication systems. In this example, plays the role of the number
of users in the system, and is the length of the coding sequence of each of
the users. Each user transmits at the same time and uses the same frequency,
and the codes are used to distinguish the signals of the separate users. The
results imply large deviation bounds for the probability of a bit error due to
the interference of the various users.Comment: corrected some typing errors, and extended Theorem 3.1 to Wishart
matrices; to appear in Advances of Applied Probabilit
The winner takes it all
We study competing first passage percolation on graphs generated by the
configuration model. At time 0, vertex 1 and vertex 2 are infected with the
type 1 and the type 2 infection, respectively, and an uninfected vertex then
becomes type 1 (2) infected at rate () times the number
of edges connecting it to a type 1 (2) infected neighbor. Our main result is
that, if the degree distribution is a power-law with exponent ,
then, as the number of vertices tends to infinity and with high probability,
one of the infection types will occupy all but a finite number of vertices.
Furthermore, which one of the infections wins is random and both infections
have a positive probability of winning regardless of the values of
and . The picture is similar with multiple starting points for the
infections
Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: I. The higher-point functions
We consider the critical spread-out contact process in Z^d with d\ge1, whose
infection range is denoted by L\ge1. In this paper, we investigate the r-point
function \tau_{\vec t}^{(r)}(\vec x) for r\ge3, which is the probability that,
for all i=1,...,r-1, the individual located at x_i\in Z^d is infected at time
t_i by the individual at the origin o\in Z^d at time 0. Together with the
results of the 2-point function in [van der Hofstad and Sakai, Electron. J.
Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially
rely, we prove that the r-point functions converge to the moment measures of
the canonical measure of super-Brownian motion above the upper-critical
dimension 4. We also prove partial results for d\le4 in a local mean-field
setting.Comment: 75 pages, 12 figure
From trees to graphs: collapsing continuous-time branching processes
Continuous-time branching processes (CTBPs) are powerful tools in random
graph theory, but are not appropriate to describe real-world networks, since
they produce trees rather than (multi)graphs. In this paper we analyze
collapsed branching processes (CBPs), obtained by a collapsing procedure on
CTBPs, in order to define multigraphs where vertices have fixed out-degree
. A key example consists of preferential attachment models (PAMs), as
well as generalized PAMs where vertices are chosen according to their degree
and age. We identify the degree distribution of CBPs, showing that it is
closely related to the limiting distribution of the CTBP before collapsing. In
particular, this is the first time that CTBPs are used to investigate the
degree distribution of PAMs beyond the tree setting.Comment: 18 pages, 3 figure
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