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Remco van der Hofstad

Publications and source records attributed to Remco van der Hofstad.

At least 109 records · Page 6Linked to original sources
arXiv2017

The dynamics of power laws: Fitness and aging in preferential attachment trees

Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment models in which the birth rates are linear functions. We are motivated by citation networks, where power-law citation counts are observed as well as aging in the citation patterns. To model this, we introduce fitness and age-dependence in these birth processes. The multiplicative fitness moderates the rate at which children are born, while the aging is integrable, so that individuals receives a finite number of children in their lifetime. We show the existence of a limiting degree distribution for such processes. In the preferential attachment case, where fitness and aging are absent, this limiting degree distribution is known to have power-law tails. We show that the limiting degree distribution has exponential tails for bounded fitnesses in the presence of integrable aging, while the power-law tail is restored when integrable aging is combined with fitness with unbounded support with at most exponential tails. In the absence of integrable aging, such processes are explosive.

math.PR↗
arXiv2017

Triadic closure in configuration models with unbounded degree fluctuations

The configuration model generates random graphs with any given degree distribution, and thus serves as a null model for scale-free networks with power-law degrees and unbounded degree fluctuations. For this setting, we study the local clustering $c(k)$, i.e., the probability that two neighbors of a degree-$k$ node are neighbors themselves. We show that $ c(k)$ progressively falls off with $k$ and eventually for $k=Ω(\sqrt{n})$ settles on a power law $c(k)\sim k^{-2(3-τ)}$ with $τ\in(2,3)$ the power-law exponent of the degree distribution. This fall-off has been observed in the majority of real-world networks and signals the presence of modular or hierarchical structure. Our results agree with recent results for the hidden-variable model and also give the expected number of triangles in the configuration model when counting triangles only once despite the presence of multi-edges. We show that only triangles consisting of triplets with uniquely specified degrees contribute to the triangle counting.

math.PR↗
arXiv2017

Clustering Spectrum of scale-free networks

Real-world networks often have power-law degrees and scale-free properties such as ultra-small distances and ultra-fast information spreading. In this paper, we study a third universal property: three-point correlations that suppress the creation of triangles and signal the presence of hierarchy. We quantify this property in terms of $\bar c(k)$, the probability that two neighbors of a degree-$k$ node are neighbors themselves. We investigate how the clustering spectrum $k\mapsto\bar c(k)$ scales with $k$ in the hidden variable model and show that $c(k)$ follows a {\it universal curve} that consists of three $k$-ranges where $\bar c(k)$ remains flat, starts declining, and eventually settles on a power law $\bar c(k)\sim k^{-α}$ with $α$ depending on the power law of the degree distribution. We test these results against ten contemporary real-world networks and explain analytically why the universal curve properties only reveal themselves in large networks.

cs.SI↗
arXiv2017

When is a scale-free graph ultra-small?

In this paper we study typical distances in the configuration model, when the degrees have asymptotically infinite variance. We assume that the empirical degree distribution follows a power law with exponent $τ\in (2,3)$, up to value $n^{β_n}$ for some $β_n\gg (\log n)^{-γ}$ and $γ\in(0,1)$. This assumption is satisfied for power law i.i.d. degrees, and also includes truncated power-law distributions where the (possibly exponential) truncation happens at $n^{β_n}$. We show that the graph distance between two uniformly chosen vertices centers around $2 \log \log (n^{β_n}) / |\log (τ-2)| + 1/(β_n(3-τ))$, with tight fluctuations. Thus, the graph is an \emph{ultrasmall world} whenever $1/β_n=o(\log\log n)$. We determine the distribution of the fluctuations around this value, in particular we prove that these are non-converging tight random variables that show $\log \log$-periodicity. We describe the topology and number of shortest paths: We show that the number of shortest paths is of order $n^{f_nβ_n}$, where $f_n \in (0,1)$ is a random variable that oscillates with $n$. The two end-segments of any shortest path have length $\log \log (n^{β_n}) / |\log (τ-2)|$+tight, and the total degree is increasing towards the middle of the path on these segments. The connecting middle segment has length $1/(β_n(3-τ))$+tight, and it contains only vertices with degree at least of order $n^{(1-f_n)β_n}$, thus all the degrees on this segment are comparable to the maximal degree. Our theorems also apply when instead of truncating the degrees, we start with a configuration model and we remove every vertex with degree at least $n^{β_n}$, and the edges attached to these vertices. This sheds light on the attack vulnerability of the configuration model with infinite variance degrees.

math.PR↗
arXiv2017

Consistent Estimation in General Sublinear Preferential Attachment Trees

We propose an empirical estimator of the preferential attachment function $f$ in the setting of general preferential attachment trees. Using a supercritical continuous-time branching process framework, we prove the almost sure consistency of the proposed estimator. We perform simulations to study the empirical properties of our estimators.

math.ST↗
arXiv2017

Backbone scaling limit of the high-dimensional IIC: extended version

We identify the scaling limit of the backbone of the high-dimensional incipient infinite cluster (IIC), both in the finite-range and the long-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 that keeps track of the number of pivotal bonds.

math.PR↗
arXiv2017

Big jobs arrive early: From critical queues to random graphs

We consider a queue to which only a finite pool of $n$ customers can arrive, at times depending on their service requirement. A customer with stochastic service requirement $S$ arrives to the queue after an exponentially distributed time with mean $S^{-α}$ for some $α\in[0,1]$; so larger service requirements trigger customers to join earlier. This finite-pool queue interpolates between two previously studied cases: $α= 0$ gives the so-called $Δ_{(i)}/G/1$ queue and $α= 1$ is closely related to the exploration process for inhomogeneous random graphs. We consider the asymptotic regime in which the pool size $n$ grows to infinity and establish that the scaled queue-length process converges to a diffusion process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of activity. We also describe how this first busy period of the queue gives rise to a critically connected random forest.

math.PR↗
arXiv2017

Mean-field behavior for nearest-neighbor percolation in $d>10$

We prove that nearest-neighbor percolation in dimensions $d\geq 11$ displays mean-field behavior by proving that the infrared bound holds, in turn implying the finiteness of the percolation triangle diagram. The finiteness of the triangle implies the existence and mean-field values of various critical exponents, such as $γ=1, β=1, δ=2$. We also prove sharp $x$-space asymptotics for the two-point function and the existence of various arm exponents. Such results had previously been obtained in unpublished work by Hara and Slade for nearest-neighbor percolation in dimension $d\geq 19$, so that we bring the dimension above which mean-field behavior is rigorously proved down from $19$ to $11$. Our results also imply sharp bounds on the critical value of nearest-neighbor percolation on $\mathbb{Z}^d$, which are provably at most $1.306\%$ off in $d=11$. We make use of the general method analyzed in the accompanying paper "Generalized approach to the non-backtracking lace expansion" by Fitzner and van der Hofstad, which proposes to use a lace expansion perturbing around non-backtracking random walk. This proof is {\em computer-assisted}, relying on (1) rigorous numerical upper bounds on various simple random walk integrals as proved by Hara and Slade (1992) and (2) a verification that the derived numerical conditions hold true. These two ingredients are implemented in two Mathematica notebooks that can be downloaded from the website of the first author. The main steps of this paper are (a) to derive a non-backtracking lace expansion for the percolation two-point function; (b) to bound the non-backtracking lace expansion coefficients, thus showing that the general methodology applies, and (c) to describe the numerical bounds on the coefficients. In the appendix of this extended version, we give additional details about the bounds that are not given in the article version.

math.PR↗
arXiv2017

Critical window for the configuration model: finite third moment degrees

We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the asymptotic degree distribution is enough to guarantee that the sizes of the largest connected components are of the order $n^{2/3}$ and the re-scaled component sizes (ordered in a decreasing manner) converge to the ordered excursion lengths of an inhomogeneous Brownian Motion with a parabolic drift. We use percolation to study the evolution of these component sizes while passing through the critical window and show that the vector of percolation cluster-sizes, considered as a process in the critical window, converge to the multiplicative coalescent process in the sense of finite dimensional distributions. This behavior was first observed for Erdős-Rényi random graphs by Aldous (1997) and our results provide support for the empirical evidences that the nature of the phase transition for a wide array of random-graph models are universal in nature. Further, we show that the re-scaled component sizes and surplus edges converge jointly under a strong topology, at each fixed location of the scaling window.

math.PR↗
arXiv2017

Limit laws for self-loops and multiple edges in the configuration model

We consider self-loops and multiple edges in the configuration model as the size of the graph tends to infinity. The interest in these random variables is due to the fact that the configuration model, conditioned on being simple, is a uniform random graph with prescribed degrees. Simplicity corresponds to the absence of self-loops and multiple edges. We show that the number of self-loops and multiple edges converges in distribution to two independent Poisson random variables when the second moment of the empirical degree distribution converges. We also provide an estimate on the total variation distance between the number of self-loops and multiple edges and the Poisson limit of their sum. This revisits previous works of Bollobás, of Janson, of Wormald and others. The error estimates also imply sharp asymptotics for the number of simple graphs with prescribed degrees. The error estimates follow from an application of Stein's method for Poisson convergence, which is a novel method for this problem. The asymptotic independence of self-loops and multiple edges follows from a Poisson version of the Cramér-Wold device using thinning, which is of independent interest. When the degree distribution has infinite second moment, our general results break down. We can, however, prove a central limit theorem for the number of self-loops, and for the multiple edges between vertices of degrees much smaller than the square root of the size of the graph, or when we truncate the degrees similarly. Our results and proofs easily extend to directed and bipartite configuration models.

math.PR↗
arXiv2017

The multiplicative coalescent, inhomogeneous continuum random trees, and new universality classes for critical random graphs

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [23, 24, 28, 63] is as follows: for a wide array of random graph models with degree exponent $τ\in (3,4)$, distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $n^{(τ-3)/(τ-1)}$. In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of Lévy processes "without replacement" [10], yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $τ\in (3,4)$ where edges are rescaled by $n^{-(τ-3)/(τ-1)}$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact "tree-like" random fractals with finite fractal dimensions and with a dense collection of hubs (infinite degree vertices) a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal $(τ-2)/(τ-3)$ a.s., in stark contrast to the Erdős-Rényi scaling limit whose Minkowski dimension is 2 a.s. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $τ\in (3,4)$.

math.PR↗
arXiv2017

Expansion of percolation critical points for Hamming graphs

The Hamming graph $H(d,n)$ is the Cartesian product of $d$ complete graphs on $n$ vertices. Let $m=d(n-1)$ be the degree and $V = n^d$ be the number of vertices of $H(d,n)$. Let $p_c^{(d)}$ be the critical point for bond percolation on $H(d,n)$. We show that, for $d \in \mathbb N$ fixed and $n \to \infty$, \begin{equation*} p_c^{(d)}= \dfrac{1}{m} + \dfrac{2d^2-1}{2(d-1)^2}\dfrac{1}{m^2} + O(m^{-3}) + O(m^{-1}V^{-1/3}), \end{equation*} which extends the asymptotics found in \cite{BorChaHofSlaSpe05b} by one order. The term $O(m^{-1}V^{-1/3})$ is the width of the critical window. For $d=4,5,6$ we have $m^{-3} = O(m^{-1}V^{-1/3})$, and so the above formula represents the full asymptotic expansion of $p_c^{(d)}$. In \cite{FedHofHolHul16a} \st{we show that} this formula is a crucial ingredient in the study of critical bond percolation on $H(d,n)$ for $d=2,3,4$. The proof uses a lace expansion for the upper bound and a novel comparison with a branching random walk for the lower bound. The proof of the lower bound also yields a refined asymptotics for the susceptibility of a subcritical Erdős-Rényi random graph.

math.PR↗
arXiv2016

Hierarchical Configuration Model

We introduce a class of random graphs with a community structure, which we call the hierarchical configuration model. On the inter-community level, the graph is a configuration model, and on the intra-community level, every vertex in the configuration model is replaced by a community: i.e., a small graph. These communities may have any shape, as long as they are connected. For these hierarchical graphs, we find the size of the largest component, the degree distribution and the clustering coefficient. Furthermore, we determine the conditions under which a giant percolation cluster exists, and find its size.

math.PR↗
arXiv2016

Generalized approach to the non-backtracking lace expansion

The lace expansion is a powerful perturbative technique to analyze the critical behavior of random spatial processes such as the self-avoiding walk, percolation and lattice trees and animals. The non-backtracking lace expansion (NoBLE) is a modification that allows us to improve its applicability in the nearest-neighbor setting on the $\mathbb Z^d$-lattice for percolation, lattice trees and lattice animals. The NoBLE gives rise to a recursive formula that we study in this paper at a general level. We state assumptions that guarantee that the solution of this recursive formula satisfies the infrared bound. In two related papers, we show that these conditions are satisfied for percolation in $d\geq 11$, for lattice trees in $d\geq 16$ and for lattice animals in $d\geq 21$.

math.PR↗
arXiv2016

Mesoscopic scales in hierarchical configuration models

To understand mesoscopic scaling in networks, we study the hierarchical configuration model (HCM), a random graph model with community structure. The connections between the communities are formed as in a configuration model. We study the component sizes of the hierarchical configuration model at criticality when the inter-community degrees have a finite third moment. We find the conditions on the community sizes such that the critical component sizes of the HCM behave similarly as in the configuration model. Furthermore, we study critical bond percolation on the HCM. We show that the ordered components of a critical HCM on $N$ vertices are of sizes $O(N^{2/3})$. More specifically, the rescaled component sizes converge to the excursions of a Brownian motion with parabolic drift, as for the scaling limit for the configuration model under a finite third moment condition.

math.PR↗
arXiv2016

Epidemic spreading on complex networks with community structures

Many real-world networks display a community structure. We study two random graph models that create a network with similar community structure as a given network. One model preserves the exact community structure of the original network, while the other model only preserves the set of communities and the vertex degrees. These models show that community structure is an important determinant of the behavior of percolation processes on networks, such as information diffusion or virus spreading: the community structure can both \textit{enforce} as well as \textit{inhibit} diffusion processes. Our models further show that it is the mesoscopic set of communities that matters. The exact internal structures of communities barely influence the behavior of percolation processes across networks. This insensitivity is likely due to the relative denseness of the communities.

physics.soc-ph↗
arXiv2016

Local clustering in scale-free networks with hidden variables

We investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices. The class rules out self-loops and multiple edges and allows for negative degree correlations (disassortative mixing) due to infinite-variance degrees controlled by a structural cutoff $h_s$ and natural cutoff $h_c$. We show that local clustering decreases with the hidden variable (or degree). We also determine how the average clustering coefficient $C$ scales with the network size $N$, as a function of $h_s$ and $h_c$. For scale-free networks with exponent $2<τ<3$ and the default choices $h_s\sim N^{1/2}$ and $h_c\sim N^{1/(τ-1)}$ this gives $C\sim N^{2-τ}\ln N$ for the universality class at hand. We characterize the extremely slow decay of $C$ when $τ\approx 2$ and show that for $τ=2.1$, say, clustering only starts to vanish for networks as large as $N=10^{11}$.

math.PR↗