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 $\Z^d$ 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

Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems

We study sample covariance matrices of the form $W=\frac 1n C C^T$, where $C$ is a $k\times n$ matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of $C$ are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when $k$ is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of $W$ when either $k$ is fixed and $n\to \infty$, or $k_n\to \infty$ with $k_n=o(n/\log\log{n})$, 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 $k$ large are for the case where the entries of $C$ are $\pm1$ 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 $C$. 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, $k$ plays the role of the number of users in the system, and $n$ 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

Diameter of the stochastic mean-field model of distance

We consider the complete graph \cK_n on $n$ vertices with exponential mean $n$ edge lengths. Writing $C_{ij}$ for the weight of the smallest-weight path between vertex $i,j\in [n]$, Janson showed that $\max_{i,j\in [n]} C_{ij}/\log{n}$ converges in probability to 3. We extend this result by showing that $\max_{i,j\in [n]} C_{ij} - 3\log{n}$ 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

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 $\lambda_1$ ($\lambda_2$) 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 $\tau\in(2,3)$, 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 $\lambda_1$ and $\lambda_2$. 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 $m\geq 2$. 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

Random graph asymptotics on high-dimensional tori

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d>6 for sufficient spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times $V^{2/3}$ and below by a small constant times $V^{2/3}(log V)^{-4/3}$, where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on Z^d under which the lower bound can be improved to small constant times $V^{2/3}$, i.e., we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by Aizenman (1997), apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. Our method is crucially based on the results by Borgs, Chayes, van der Hofstad, Slade and Spencer (2005a, 2005b), where the $V^{2/3}$ scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on Z^d. We also strongly rely on mean-field results for percolation on Z^d proved by Hara (1990, 2005), Hara and Slade (1990) and Hara, van der Hofstad and Slade (2003).Comment: 22 page