arXiv subjects

Remco van der Hofstad

Publications and source records attributed to Remco van der Hofstad.

At least 145 records · Page 8Linked to original sources

arXiv2014

Degree-degree correlations in random graphs with heavy-tailed degrees

Mixing patterns in large self-organizing networks, such as the Internet, the World Wide Web, social and biological networks are often characterized by degree-degree {dependencies} between neighbouring nodes. One of the problems with the commonly used Pearson's correlation coefficient (termed as the assortativity coefficient) is that {in disassortative networks its magnitude decreases} with the network size. This makes it impossible to compare mixing patterns, for example, in two web crawls of different size. We start with a simple model of two heavy-tailed highly correlated random variable $X$ and $Y$, and show that the sample correlation coefficient converges in distribution either to a proper random variable on $[-1,1]$, or to zero, and if $X,Y\ge 0$ then the limit is non-negative. We next show that it is non-negative in the large graph limit when the degree distribution has an infinite third moment. We consider the alternative degree-degree dependency measure, based on the Spearman's rho, and prove that it converges to an appropriate limit under very general conditions. We verify that these conditions hold in common network models, such as configuration model and Preferential Attachment model. We conclude that rank correlations provide a suitable and informative method for uncovering network mixing patterns.

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arXiv2014

The front of the epidemic spread and first passage percolation

In this paper we establish a connection between epidemic models on random networks with general infection times considered in Barbour and Reinert 2013 and first passage percolation. Using techniques developed in Bhamidi, van der Hofstad, Hooghiemstra 2012, when each vertex has infinite contagious periods, we extend results on the epidemic curve in Barbour Reinert 2013 from bounded degree graphs to general sparse random graphs with degrees having finite third moments as the number of vertices tends to infinity. We also study the epidemic trail between the source and typical vertices in the graph. This connection to first passage percolation can be also be used to study epidemic models with general contagious periods as in Barbour Reinert 2013 without bounded degree assumptions.

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arXiv2013

Random walk on the high-dimensional IIC

We study the asymptotic behavior the exit times of random walk from Euclidean balls around the origin of the incipient infinite cluster in a manner inspired by [26]. We do this by obtaining bounds on the effective resistance between the origin and the boundary of these Euclidean balls. We show that the geometric properties of long-range percolation clusters are significantly different from those of finite-range clusters. We also study the behavior of random walk on the backbone of the IIC and we prove that the Alexander-Orbach conjecture holds for the incipient infinite cluster in high dimensions, both for long-range percolation and for finite-range percolation.

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arXiv2013

Backbone scaling limit of the high-dimensional IIC

We identify the scaling limit of the backbone of the high-dimensional incipient infinite cluster (IIC), both in the long- as well as in the finite-range setting. In the finite-range setting, this scaling limit is Brownian motion, in the long-range setting, it is a stable motion. The proof relies on a novel lace expansion for percolation that resembles the original expansion for self-avoiding walks by Brydges and Spencer in 1985. This expansion is interesting in its own right.

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arXiv2013

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 Erdős-Rényi random graph in work by Riordan and Wormald.

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arXiv2013

Cycle structure of percolation on high-dimensional tori

In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one for percolation on the complete graph or Erdos-Renyi random graph, raising the question whether the scaling limits of the largest connected components, as identified by Aldous (1997), are also equal. In this paper, we investigate the cycle structure of the largest critical components for high-dimensional percolation on the torus (Z/rZ)^d. While percolation clusters naturally have many short cycles, we show that the long cycles, i.e., cycles that pass through the boundary of the cube of width r/4 centered around each of their vertices, have length of order r^{d/3}, as on the critical Erdos-Renyi random graph. On the Erdos-Renyi random graph, cycles play an essential role in the scaling limit of the large critical clusters, as identified by Addario-Berry, Broutin and Goldschmidt (arXiv:0908.3629). Our proofs crucially rely on various new estimates of probabilities of the existence of open paths in critical Bernoulli percolation on Z^d with constraints on their lengths. We believe these estimates are interesting in their own right.

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arXiv2013

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.

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arXiv2012

Non-backtracking random walk

We consider non-backtracking random walk (NBW) in the nearest-neighbor setting on the Zd-lattice and on tori. We evaluate the eigensystem of the m X m-dimensional transition matrix of NBW where m denote the degree of the graph. We use its eigensystem to show a functional central limit theorem for NBW on Zd and to obtain estimates on the convergence towards the stationary distribution for NBW on the torus.

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arXiv2012

Novel scaling limits for critical inhomogeneous random graphs

We find scaling limits for the sizes of the largest components at criticality for rank-1 inhomogeneous random graphs with power-law degrees with power-law exponent τ. We investigate the case where $τ\in(3,4)$, so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by $n^{-(τ-2)/(τ-1)}$, converge to hitting times of a "thinned" Lévy process, a special case of the general multiplicative coalescents studied by Aldous [Ann. Probab. 25 (1997) 812-854] and Aldous and Limic [Electron. J. Probab. 3 (1998) 1-59]. Our results should be contrasted to the case τ>4, so that the third moment is finite. There, instead, the sizes of the components rescaled by $n^{-2/3}$ converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in Aldous [Ann. Probab. 25 (1997) 812-854] for the Erdős-Rényi random graph and extended to the present setting in Bhamidi, van der Hofstad and van Leeuwaarden [Electron. J. Probab. 15 (2010) 1682-1703] and Turova [(2009) Preprint].

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arXiv2012

Short paths for first passage percolation on the complete graph

We study the complete graph equipped with a topology induced by independent and identically distributed edge weights. The focus of our analysis is on the weight W_n and the number of edges H_n of the minimal weight path between two distinct vertices in the weak disorder regime. We establish novel and simple first and second moment methods using path counting to derive first order asymptotics for the considered quantities. Our results are stated in terms of a sequence of parameters (s_n) that quantifies the extreme-value behaviour of the edge weights, and that describes different universality classes for first passage percolation on the complete graph. These classes contain both n-independent and n-dependent edge weight distributions. The method is most effective for the universality class containing the edge weights E^{s_n}, where E is an exponential(1) random variable and s_n log n -> infty, s_n^2 log n -> 0. We discuss two types of examples from this class in detail. In addition, the class where s_n log n stays finite is studied. This article is a contribution to the program initiated in \cite{BhaHof12}.

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arXiv2012

Ising critical exponents on random trees and graphs

We study the critical behavior of the ferromagnetic Ising model on random trees as well as so-called locally tree-like random graphs. We pay special attention to trees and graphs with a power-law offspring or degree distribution whose tail behavior is characterized by its power-law exponent $τ>2$. We show that the critical temperature of the Ising model equals the inverse hyperbolic tangent of the inverse of the mean offspring or mean forward degree distribution. In particular, the inverse critical temperature equals zero when $τ\in(2,3]$ where this mean equals infinity. We further study the critical exponents $δ, β$ and $γ$, describing how the (root) magnetization behaves close to criticality. We rigorously identify these critical exponents and show that they take the values as predicted by Dorogovstev, et al. and Leone et al. These values depend on the power-law exponent $τ$, taking the mean-field values for $τ>5$, but different values for $τ\in(3,5)$.

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arXiv2012

Unlacing the lace expansion: a survey to hypercube percolation

The purpose of this note is twofold. First, we survey the study of the percolation phase transition on the Hamming hypercube {0,1}^m obtained in the series of papers [9,10,11,24]. Secondly, we explain how this study can be performed without the use of the so-called "lace-expansion" technique. To that aim, we provide a novel simple proof that the triangle condition holds at the critical probability. We hope that some of these techniques will be useful to obtain non-perturbative proofs in the analogous, yet much more difficult study on high-dimensional tori.

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arXiv2012

Universality for first passage percolation on sparse random graphs

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^2\log{X}-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path or hopcount. The hopcount satisfies a central limit theorem where the norming constants are expressible in terms of the parameters of an associated continuous-time branching process. Centered by a multiple of \log{n}, where the constant is the inverse of the Malthusian rate of growth of the associated branching process, the minimal weight converges in distribution. The limiting random variable equals the sum of the logarithms of the martingale limits of the branching processes that measure the relative growth of neighborhoods about the two vertices, and a Gumbel random variable, and thus shows a remarkably universal behavior. The proofs rely on a refined coupling between the shortest path problems on these graphs and continuous-time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination. The results extend to a host of related random graph models, ranging from random r-regular graphs, inhomogeneous random graphs and uniform random graphs with a prescribed degree sequence.

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arXiv2012

High-dimensional incipient infinite clusters revisited

The incipient infinite cluster (IIC) measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite. Using the lace expansion, we construct the IIC measure for high-dimensional percolation models in three different ways, extending previous work by the second author and Jarai. We show that each construction yields the same measure, indicating that the IIC is a robust object. Furthermore, our constructions apply to spread-out versions of both finite-range and long-range percolation models. We also obtain estimates on structural properties of the IIC, such as the volume of the intersection between the IIC and Euclidean balls.

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arXiv2012

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

In the recent past, there has been a concerted effort to develop mathematical models for real-world networks and to analyze various dynamics on these models. One particular problem of significant importance is to understand the effect of random edge lengths or costs on the geometry and flow transporting properties of the network. Two different regimes are of great interest, the weak disorder regime where optimality of a path is determined by the sum of edge weights on the path and the strong disorder regime where optimality of a path is determined by the maximal edge weight on the path. In the context of the stochastic mean-field model of distance, we provide the first mathematically tractable model of weak disorder and show that no transition occurs at finite temperature. Indeed, we show that for every finite temperature, the number of edges on the minimal weight path (i.e., the hopcount) is $Θ(\log{n})$ and satisfies a central limit theorem with asymptotic means and variances of order $Θ(\log{n})$, with limiting constants expressible in terms of the Malthusian rate of growth and the mean of the stable-age distribution of an associated continuous-time branching process. More precisely, we take independent and identically distributed edge weights with distribution $E^s$ for some parameter $s>0$, where $E$ is an exponential random variable with mean 1. Then the asymptotic mean and variance of the central limit theorem for the hopcount are $s\log{n}$ and $s^2\log{n}$, respectively. We also find limiting distributional asymptotics for the value of the minimal weight path in terms of extreme value distributions and martingale limits of branching processes.

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arXiv2012

Hypercube percolation

We study bond percolation on the Hamming hypercube {0,1}^m around the critical probability p_c. It is known that if p=p_c(1+O(2^{-m/3})), then with high probability the largest connected component C_1 is of size Theta(2^{2m/3}) and that this quantity is non-concentrated. Here we show that for any sequence eps_m such that eps_m=o(1) but eps_m >> 2^{-m/3} percolation on the hypercube at p_c(1+eps_m) has |C_1| = (2+o(1)) eps_m 2^m and |C_2| = o(eps_m 2^m) with high probability, where C_2 is the second largest component. This resolves a conjecture of Borgs, Chayes, the first author, Slade and Spencer [17].

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arXiv2011

The survival probability and r-point functions in high dimensions

In this paper we investigate the survival probability, θ_n, in high-dimensional statistical physical models, where θ_n denotes the probability that the model survives up to time n. We prove that if the r-point functions scale to those of the canonical measure of super-Brownian motion, and if a certain self-repellence condition is satisfied, then nθ_n\ra 2/(AV), where A is the asymptotic expected number of particles alive at time n, and V is the vertex factor of the model. Our results apply to spread-out lattice trees above 8 dimensions, spread-out oriented percolation above 4+1 dimensions, and the spread-out contact process above 4+1 dimensions. In the case of oriented percolation, this reproves a result by the first author, den Hollander and Slade (that was proved using heavy lace expansion arguments), at the cost of losing explicit error estimates. We further derive several consequences of our result involving the scaling limit of the number of particles alive at time proportional to n. Our proofs are based on simple weak convergence arguments.

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arXiv2011

Scale-free percolation

We formulate and study a model for inhomogeneous long-range percolation on $\Zbold^d$. Each vertex $x\in\Zbold^d$ is assigned a non-negative weight $W_x$, where $(W_x)_{x\in\Zbold^d}$ are i.i.d.\ random variables. Conditionally on the weights, and given two parameters $α,λ>0$, the edges are independent and the probability that there is an edge between $x$ and $y$ is given by $p_{xy}=1-\exp\{-λW_xW_y/|x-y|^α\}$. The parameter $λ$ is the percolation parameter, while $α$ describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of $W_x$ is regularly varying with exponent $τ-1$, then the tail of the degree distribution is regularly varying with exponent $γ=α(τ-1)/d$. The parameter $γ$ turns out to be crucial for the behavior of the model. Conditions on the weight distribution and $γ$ are formulated for the existence of a critical value $λ_c\in(0,\infty)$ such that the graph contains an infinite component when $λ>λ_c$ and no infinite component when $λ<λ_c$. Furthermore, a phase transition is established for the graph distances between vertices in the infinite component at the point $γ=2$, that is, at the point where the degrees switch from having finite to infinite second moment. The model can be viewed as an interpolation between long-range percolation and models for inhomogeneous random graphs, and we show that the behavior shares the interesting features of both these models.

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