arXiv subjects

Remco van der Hofstad

Publications and source records attributed to Remco van der Hofstad.

At least 127 records · Page 7Linked to original sources

arXiv2016

Component structure of the configuration model: barely supercritical case

We study near-critical behavior in the configuration model. Let $D_n$ be the degree of a random vertex. We let $ν_n={\mathbb E} [D_n(D_n-1)]/{\mathbb E}[D_n]$ and, assuming that $ν_n \to 1$ as $n \to \infty$, we write $\varepsilon_n=ν_n-1$. We call the setting where $\varepsilon_n n^{1/3}/({\mathbb E}[D_n^3])^{2/3} \to \infty$ the {\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \to \infty$. Let $D_n^*$ denote the size-biased version of $D_n$. We prove that there is a unique giant component of size $n ρ_n {\mathbb E} D_n (1+o(1))$, where $ρ_n$ denotes the survival probability of a branching process with offspring distribution $D_n^*-1$. This extends earlier results of Janson and Luczak~\cite{JanLuc07}, as well as those of Janson, Luczak, Windridge and House~\cite{SJ300} to the case where the third moment of $D_n$ is unbounded, filling the gap in the literature. We further study the size of the largest component in the \emph{critical} regime, where $\varepsilon_n = O(n^{-1/3} ({\mathbb E} D_n^3)^{2/3})$, extending and complementing results of Hatami and Molloy~\cite{HatamiMolloy}.

math.PR↗

arXiv2016

Tight fluctuations of weight-distances in random graphs with infinite-variance degrees

We prove results for first-passage percolation on the configuration model with i.i.d. degrees having finite mean, infinite variance and i.i.d. weights with strictly positive support of the form Y=a+X, where a is a positive constant. We prove that the weight of the optimal path has tight fluctuations around the asymptotical mean of the graph-distance if and only if the following condition holds: the random variable X is such that the continuous-time branching process describing first-passage percolation exploration in the same graph with excess edge weight X has a positive probability to reach infinitely many individuals in a finite time. This shows that almost shortest paths in the graph-distance proliferate, in the sense that there are even ones having tight total excess edge weight for various edge-weight distributions.

math.PR↗

arXiv2016

Ising critical behavior of inhomogeneous Curie-Weiss models and annealed random graphs

We study the critical behavior for inhomogeneous versions of the Curie-Weiss model, where the coupling constant $J_{ij}(β)$ for the edge $ij$ on the complete graph is given by $J_{ij}(β)=βw_iw_j/(\sum_{k\in[N]}w_k)$. We call the product form of these couplings the rank-1 inhomogeneous Curie-Weiss model. This model also arises (with inverse temperature $β$ replaced by $\sinh(β)$) from the annealed Ising model on the generalized random graph. We assume that the vertex weights $(w_i)_{i\in[N]}$ are regular, in the sense that their empirical distribution converges and the second moment converges as well. We identify the critical temperatures and exponents for these models, as well as a non-classical limit theorem for the total spin at the critical point. These depend sensitively on the number of finite moments of the weight distribution. When the fourth moment of the weight distribution converges, then the critical behavior is the same as on the (homogeneous) Curie-Weiss model, so that the inhomogeneity is weak. When the fourth moment of the weights converges to infinity, and the weights satisfy an asymptotic power law with exponent $τ$ with $τ\in(3,5)$, then the critical exponents depend sensitively on $τ$. In addition, at criticality, the total spin $S_{N}$ satisfies that $S_{N}/N^{(τ-1)/(τ-2)}$ converges in law to some limiting random variable whose distribution we explicitly characterize.

math.PR↗

arXiv2016

Finite-pool queues with heavy-tailed services

We consider the $Δ_{(i)}/G/1$ queue, in which a a total of $n$ customers independently demand service after an exponential time. We focus on the case of heavy-tailed service times, and assume that the tail of the service time distribution decays like $x^{-α}$, with $α\in (1,2)$. We consider the asymptotic regime in which the population size grows to infinity and establish that the scaled queue length process converges to an α-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the headstart that is needed to create a long period of activity. This result should be contrasted with the case of light-tailed service times, which was shown to have a similar scaling limit, but then with a Brownian motion instead of an α-stable process.

math.PR↗

arXiv2016

Lace expansion for dummies

We show Green's function asymptotic upper bound for the two-point function of weakly self-avoiding walk in dimension bigger than 4, revisiting a classic problem. Our proof relies on Banach algebras to analyse the lace-expansion fixed point equation and is simpler than previous approaches in that it avoids Fourier transforms.

math.PR↗

arXiv2016

Power-law relations in random networks with communities

Most random graph models are locally tree-like - do not contain short cycles - rendering them unfit for modeling networks with a community structure. We introduce the hierarchical configuration model (HCM), a generalization of the configuration model that includes community structures, while properties such as the size of the giant component, and the size of the giant percolating cluster under bond percolation can still be derived analytically. Viewing real-world networks as realizations of the HCM, we observe two previously unobserved power-law relations: between the number of edges inside a community and the community sizes, and between the number of edges going out of a community and the community sizes. We also relate the power-law exponent $τ$ of the degree distribution with the power-law exponent of the community size distribution $γ$. In the special case of extremely dense communities (e.g., complete graphs), this relation takes the simple form $τ=γ-1$.

physics.soc-ph↗

arXiv2016

Connectivity Threshold for random subgraphs of the Hamming graph

We study the connectivity of random subgraphs of the $d$-dimensional Hamming graph $H(d, n)$, which is the Cartesian product of $d$ complete graphs on $n$ vertices. We sample the random subgraph with an i.i.d.\ Bernoulli bond percolation on $H(d,n)$ with parameter $p$. We identify the window of the transition: when $ np- \log n \to - \infty$ the probability that the graph is connected goes to $0$, while when $ np- \log n \to + \infty$ it converges to $1$. We also investigate the connectivity probability inside the critical window, namely when $ np- \log n \to t \in \mathbb{R}$. We find that the threshold does not depend on $d$, unlike the phase transition of the giant connected component the Hamming graph (see [Bor et al, 2005]). Within the critical window, the connectivity probability does depend on d. We determine how.

math.PR↗

arXiv2016

Annealed central limit theorems for the Ising model on random graphs

The aim of this paper is to prove central limit theorems with respect to the annealed measure for the magnetization rescaled by $\sqrt{N}$ of Ising models on random graphs. More precisely, we consider the general rank-1 inhomogeneous random graph (or generalized random graph), the 2-regular configuration model and the configuration model with degrees 1 and 2. For the generalized random graph, we first show the existence of a finite annealed inverse critical temperature $0 \leq β^{\mathrm \scriptscriptstyle an}_c < \infty$ and then prove our results in the uniqueness regime, i.e., the values of inverse temperature $β$ and external magnetic field $B$ for which either $β< β^{\mathrm \scriptscriptstyle an}_c$ and $B=0$, or $β>0$ and $B \neq 0$. In the case of the configuration model, the central limit theorem holds in the whole region of the parameters $β$ and $B$, because phase transitions do not exist for these systems as they are closely related to one-dimensional Ising models. Our proofs are based on explicit computations that are possible since the Ising model on the generalized random graph in the annealed setting is reduced to an inhomogeneous Curie-Weiss model, while the analysis of the configuration model with degrees only taking values 1 and 2 relies on that of the classical one-dimensional Ising model.

math.PR↗

arXiv2016

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 $λ_1$ ($λ_2$) 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 $τ\in(2,3)$, 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 $λ_1$ and $λ_2$. The picture is similar with multiple starting points for the infections.

math.PR↗

arXiv2015

Heavy-traffic analysis through uniform acceleration of queues with diminishing populations

We consider a single server queue that serves a finite population of $n$ customers that will enter the queue (require service) only once, also known as the $Δ_{(i)}/G/1$ queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets $n$ and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over time, and hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue length process converges to a Brownian motion with parabolic drift, a stochastic-process limit that captures the effect of a diminishing population by a negative quadratic drift. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period.

math.PR↗

arXiv2015

Degree distribution of shortest path trees and bias of network sampling algorithms

In this article, we explicitly derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the asymptotics of the degree distribution for large degrees of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic mean-field model of distance), as well as on the configuration model with edge-weights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree power-law exponent. We also consider random $r$-regular graphs for large $r$, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean-field model of distance. We use our results to shed light on an empirically observed bias in network sampling methods. This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra [Ann. Appl. Probab. 20 (2010) 1907-1965], [Combin. Probab. Comput. 20 (2011) 683-707], [Adv. in Appl. Probab. 42 (2010) 706-738] of analyzing the effect of attaching random edge lengths on the geometry of random network models.

math.PR↗

arXiv2015

First passage percolation on random graphs with infinite variance degrees

We prove non-universality results for first-passage percolation on the configuration model with i.i.d. degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviors are possible. When the weights are a.s. larger than a constant, the weight and number of edges in the graph grow proportionally to loglog(n), as for the graph distances. On the other hand, when the continuous-time branching process describing the first passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the processes started from the two sources. This non-universality is in sharp contrast to the setting where the degree sequence has a finite variance (see Bhamidi, Hofstad and Hooghiemstra arXiv: 1210.6839).

math.PR↗

arXiv2015

Fixed speed competition on the configuration model with infinite variance degrees: equal speeds

We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following a power-law distribution with exponent $τ\in (2,3)$. In this model two colors spread with a fixed and equal speed on the unweighted random graph. We analyse how many vertices the two colors paint eventually. We show that coexistence sensitively depends on the initial local neighborhoods of the source vertices: if these neighborhoods are `dissimilar enough', then there is no coexistence, and the `loser' color paints a polynomial fraction of the vertices with a random exponent. If the local neighborhoods of the starting vertices are `similar enough', then there is coexistence, i.e., both colors paint a strictly positive proportion of vertices. We give a quantitative characterization of `similar' local neighborhoods: two random variables describing the double exponential growth of local neighborhoods of the source vertices must be within a factor $τ-2$ of each other. Both of the two outcomes happen with positive probability with asymptotic value that is explicitly computable. This picture reinforces the common belief that location is an important feature in advertising. This paper is a follow-up of the similarly named paper that handles the case when the speeds of the two colors are not equal. There, we have shown that the faster color paints almost all vertices, while the slower color paints only a random sub-polynomial fraction of the vertices.

math.PR↗

arXiv2014

Quenched central limit theorems for the Ising model on random graphs

The main goal of the paper is to prove central limit theorems for the magnetization rescaled by $\sqrt{N}$ for the Ising model on random graphs with $N$ vertices. Both random quenched and averaged quenched measures are considered. We work in the uniqueness regime $β>β_c$ or $β>0$ and $B\neq0$, where $β$ is the inverse temperature, $β_c$ is the critical inverse temperature and $B$ is the external magnetic field. In the random quenched setting our results apply to general tree-like random graphs (as introduced by Dembo, Montanari and further studied by Dommers and the first and third author) and our proof follows that of Ellis in $\mathbb{Z}^d$. For the averaged quenched setting, we specialize to two particular random graph models, namely the 2-regular configuration model and the configuration model with degrees 1 and 2. In these cases our proofs are based on explicit computations relying on the solution of the one dimensional Ising models.

math.PR↗

arXiv2014

Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds

We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following a power-law distribution with exponent tau in (2,3). In this model two colors spread with a fixed but not necessarily equal speed on the unweighted random graph. We show that if the speeds are not equal, then the faster color paints almost all vertices, while the slower color can paint only a random subpolynomial fraction of the vertices. We investigate the case when the speeds are equal and typical distances in a follow-up paper.

math.PR↗

arXiv2014

Strongly reinforced Pólya urns with graph-based competition

We introduce a class of reinforcement models where, at each time step $t$, one first chooses a random subset $A_t$ of colours (independent of the past) from $n$ colours of balls, and then chooses a colour $i$ from this subset with probability proportional to the number of balls of colour $i$ in the urn raised to the power $α>1$. We consider stability of equilibria for such models and establish the existence of phase transitions in a number of examples, including when the colours are the edges of a graph, a context which is a toy model for the formation and reinforcement of neural connections.

math.PR↗

arXiv2014

Cluster tails for critical power-law inhomogeneous random graphs

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained (see previous work by Bhamidi, van der Hofstad and van Leeuwaarden). It was proved that when the degrees obey a power law with exponent in the interval (3,4), the sequence of clusters ordered in decreasing size and scaled appropriately converges as n goes to infinity to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u, as a function of u. This extends a related result of Pittel for the Erdős-Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

math.PR↗

arXiv2014

Large deviations for power-law thinned Levy processes

This paper deals with the large deviations behavior of a stochastic process called thinned Levy process. This process appeared recently as a stochastic-process limit in the context of critical inhomogeneous random graphs. The process has a strong negative drift, while we are interested in the rare event of the process being positive at large times. To characterize this rare event, we identify a tilted measure. This presents some challenges inherent to the power-law nature of the thinned Levy process. General principles prescribe that the tilt should follow from a variational problem, but in the case of the thinned Levy process this involves a Riemann sum that is hard to control. We choose to approximate the Riemann sum by its limiting integral, derive the first-order correction term, and prove that the tilt that follows from the corresponding approximate variational problem is sufficient to establish the large deviations results.

math.PR↗