arXiv subjects

Remco van der Hofstad

Publications and source records attributed to Remco van der Hofstad.

At least 91 records · Page 5Linked to original sources

arXiv2019

Scale-free Networks Well Done

We bring rigor to the vibrant activity of detecting power laws in empirical degree distributions in real-world networks. We first provide a rigorous definition of power-law distributions, equivalent to the definition of regularly varying distributions that are widely used in statistics and other fields. This definition allows the distribution to deviate from a pure power law arbitrarily but without affecting the power-law tail exponent. We then identify three estimators of these exponents that are proven to be statistically consistent -- that is, converging to the true value of the exponent for any regularly varying distribution -- and that satisfy some additional niceness requirements. In contrast to estimators that are currently popular in network science, the estimators considered here are based on fundamental results in extreme value theory, and so are the proofs of their consistency. Finally, we apply these estimators to a representative collection of synthetic and real-world data. According to their estimates, real-world scale-free networks are definitely not as rare as one would conclude based on the popular but unrealistic assumption that real-world data comes from power laws of pristine purity, void of noise and deviations.

physics.soc-ph↗

arXiv2019

Limit theorems for assortativity and clustering in null models for scale-free networks

An important problem in modeling networks is how to generate a randomly sampled graph with given degrees. A popular model is the configuration model, a network with assigned degrees and random connections. The erased configuration model is obtained when self-loops and multiple edges in the configuration model are removed. We prove an upper bound for the number of such erased edges for regularly-varying degree distributions with infinite variance, and use this result to prove central limit theorems for Pearson's correlation coefficient and the clustering coefficient in the erased configuration model. Our results explain the structural correlations in the erased configuration model and show that removing edges leads to different scaling of the clustering coefficient. We then prove that for the rank-1 inhomogeneous random graph, another null model that creates scale-free simple networks, the results for Pearson's correlation coefficient as well as for the clustering coefficient are similar to the results for the erased configuration model.

math.PR↗

arXiv2019

Hydra: A Multiple Blockchain Protocol for Improving Transaction Throughput

Improving transaction throughput is one of the main challenges in decentralized payment systems. Attempts to improve transaction throughput in cryptocurrencies are usually a trade-off between throughput and security or introduce a central component. We propose Hydra, a decentralized protocol that improves transaction throughput without the security trade-off and has no central component. Our novel approach distributes blocks over multiple blockchains. Hydra makes a trade-off between transaction throughput and finality, the time it takes to stabilize the record of a transaction in the blockchain. We rigorously analyze the double spend attack in a multiple-blockchain protocol. Our analysis shows that the number of transactions per second can be increased significantly while finality is within acceptable boundaries.

cs.DC↗

arXiv2019

Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time

For critical bond-percolation on high-dimensional torus, this paper proves sharp lower bounds on the size of the largest cluster, removing a logarithmic correction in the lower bound in Heydenreich and van der Hofstad (2007). This improvement finally settles a conjecture by Aizenman (1997) about the role of boundary conditions in critical high-dimensional percolation, and it is a key step in deriving further properties of critical percolation on the torus. Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on diameter and mixing time of the largest clusters. We further prove that the volume bounds apply also to any finite number of the largest clusters. The main conclusion of the paper is that the behavior of critical percolation on the high-dimensional torus is the same as for critical Erdos-Renyi random graphs. In this updated version we incorporate an erratum to be published in a forthcoming issue of Probab. Theory Relat. Fields. This results in a modification of Theorem 1.2 as well as Proposition 3.1.

math.PR↗

arXiv2019

NoBLE for lattice trees and lattice animals

We study lattice trees (LTs) and animals (LAs) on the nearest-neighbor lattice $\mathbb{Z}^d$ in high dimensions. We prove that LTs and LAs display mean-field behavior above dimension $16$ and $17$, respectively. Such results have previously been obtained by Hara and Slade in sufficiently high dimensions. The dimension above which their results apply was not yet specified. We rely on the non-backtracking lace expansion (NoBLE) method that we have recently developed. The NoBLE makes use of an alternative lace expansion for LAs and LTs that perturbs around non-backtracking random walk rather than simple random walk, leading to smaller corrections. The NoBLE method then provides a careful computational analysis that improves the dimension above which the result applies. Universality arguments predict that the upper critical dimension, above which our results apply, is equal to $d_c=8$ for both models, as is known for sufficiently spread-out models by the results of Hara and Slade mentioned earlier. The main ingredients in this paper are (a) to derive a non-backtracking lace expansion for the LT and LA two-point functions; (b) to bound the non-backtracking lace-expansion coefficients, thus showing that our general NoBLE methodology can be applied; and (c) to obtain sharp numerical bounds on the coefficients. Our proof is complemented by a computer-assisted numerical analysis that verifies that the necessary bounds used in the NoBLE are satisfied.

math-ph↗

arXiv2019

Variational principle for scale-free network motifs

For scale-free networks with degrees following a power law with an exponent $τ\in(2,3)$, the structures of motifs (small subgraphs) are not yet well understood. We introduce a method designed to identify the dominant structure of any given motif as the solution of an optimization problem. The unique optimizer describes the degrees of the vertices that together span the most likely motif, resulting in explicit asymptotic formulas for the motif count and its fluctuations. We then classify all motifs into two categories: motifs with small and large fluctuations.

cs.SI↗

arXiv2019

Cliques in rank-1 random graphs: the role of inhomogeneity

We study the asymptotic behavior of the clique number in rank-1 inhomogeneous random graphs, where edge probabilities between vertices are roughly proportional to the product of their vertex weights. We show that the clique number is concentrated on at most two consecutive integers, for which we provide an expression. Interestingly, the order of the clique number is primarily determined by the overall edge density, with the inhomogeneity only affecting multiplicative constants or adding at most a $\log\log(n)$ multiplicative factor. For sparse enough graphs the clique number is always bounded and the effect of inhomogeneity completely vanishes.

math.PR↗

arXiv2019

Large Deviations of Bivariate Gaussian Extrema

We establish sharp tail asymptotics for component-wise extreme values of bivariate Gaussian random vectors with arbitrary correlation between the components. We consider two scaling regimes for the tail event in which we demonstrate the existence of a restricted large deviations principle, and identify the unique rate function associated with these asymptotics. Our results identify when the maxima of both coordinates are typically attained by two different vs. the same index, and how this depends on the correlation between the coordinates of the bivariate Gaussian random vectors. Our results complement a growing body of work on the extremes of Gaussian processes. The results are also relevant for steady-state performance and simulation analysis of networks of infinite server queues.

math.PR↗

arXiv2019

Random walk on barely supercritical branching random walk

Let $\mathcal{T}$ be a supercritical Galton-Watson tree with a bounded offspring distribution that has mean $μ>1$, conditioned to survive. Let $φ_{\mathcal{T}}$ be a random embedding of $\mathcal{T}$ into $\mathbb{Z}^d$ according to a simple random walk step distribution. Let $\mathcal{T}_p$ be percolation on $\mathcal{T}$ with parameter $p$, and let $p_c = μ^{-1}$ be the critical percolation parameter. We consider a random walk $(X_n)_{n \ge 1}$ on $\mathcal{T}_p$ and investigate the behavior of the embedded process $φ_{\mathcal{T}_p}(X_n)$ as $n\to \infty$ and simultaneously, $\mathcal{T}_p$ becomes critical, that is, $p=p_n\searrow p_c$. We show that when we scale time by $n/(p_n-p_c)^3$ and space by $\sqrt{(p_n-p_c)/n}$, the process $(φ_{\mathcal{T}_p}(X_n))_{n \ge 1}$ converges to a $d$-dimensional Brownian motion. We argue that this scaling can be seen as an interpolation between the scaling of random walk on a static random tree and the anomalous scaling of processes in critical random environments.

math.PR↗

arXiv2018

Counting triangles in power-law uniform random graphs

We count the asymptotic number of triangles in uniform random graphs where the degree distribution follows a power law with degree exponent $τ\in(2,3)$. We also analyze the local clustering coefficient $c(k)$, the probability that two random neighbors of a vertex of degree $k$ are connected. We find that the number of triangles, as well as the local clustering coefficient, scale similarly as in the erased configuration model, where all self-loops and multiple edges of the configuration model are removed. Interestingly, uniform random graphs contain more triangles than erased configuration models with the same degree sequence. The number of triangles in uniform random graphs is closely related to that in a version of the rank-1 inhomogeneous random graph, where all vertices are equipped with weights, and the edge probabilities are moderated by asymptotically linear functions of the products of these vertex weights.

math.PR↗

arXiv2018

Scale-free network clustering in hyperbolic and other random graphs

Random graphs with power-law degrees can model scale-free networks as sparse topologies with strong degree heterogeneity. Mathematical analysis of such random graphs proved successful in explaining scale-free network properties such as resilience, navigability and small distances. We introduce a variational principle to explain how vertices tend to cluster in triangles as a function of their degrees. We apply the variational principle to the hyperbolic model that quickly gains popularity as a model for scale-free networks with latent geometries and clustering. We show that clustering in the hyperbolic model is non-vanishing and self-averaging, so that a single random graph sample is a good representation in the large-network limit. We also demonstrate the variational principle for some classical random graphs including the preferential attachment model and the configuration model.

physics.soc-ph↗

arXiv2018

Local weak convergence for PageRank

PageRank is a well-known algorithm for measuring centrality in networks. It was originally proposed by Google for ranking pages in the World-Wide Web. One of the intriguing empirical properties of PageRank is the so-called `power-law hypothesis': in a scale-free network the PageRank scores follow a power law with the same exponent as the (in-)degrees. Up to date, this hypothesis has been confirmed empirically and in several specific random graphs models. In contrast, this paper does not focus on one random graph model but investigates the existence of an asymptotic PageRank distribution, when the graph size goes to infinity, using local weak convergence. This may help to identify general network structures in which the power-law hypothesis holds. We start from the definition of local weak convergence for sequences of (random) undirected graphs, and extend this notion to directed graphs. To this end, we define an exploration process in the directed setting that keeps track of in- and out-degrees of vertices. Then we use this to prove the existence of an asymptotic PageRank distribution. As a result, the limiting distribution of PageRank can be computed directly as a function of the limiting object. We apply our results to the directed configuration model and continuous-time branching processes trees, as well as preferential attachment models.

math.PR↗

arXiv2018

Random walks on dynamic configuration models: a trichotomy

We consider a dynamic random graph on $n$ vertices that is obtained by starting from a random graph generated according to the configuration model with a prescribed degree sequence and at each unit of time randomly rewiring a fraction $α_n$ of the edges. We are interested in the mixing time of a random walk without backtracking on this dynamic random graph in the limit as $n\to\infty$, when $α_n$ is chosen such that $\lim_{n\to\infty} α_n (\log n)^2 = β\in [0,\infty]$. In [1] we found that, under mild regularity conditions on the degree sequence, the mixing time is of order $1/\sqrt{α_n}$ when $β=\infty$. In the present paper we investigate what happens when $β\in [0,\infty)$. It turns out that the mixing time is of order $\log n$, with the scaled mixing time exhibiting a one-sided cutoff when $β\in (0,\infty)$ and a two-sided cutoff when $β=0$. The occurrence of a one-sided cutoff is a rare phenomenon. In our setting it comes from a competition between the time scales of mixing on the static graph, as identified by Ben-Hamou and Salez [4], and the regeneration time of first stepping across a rewired edge.

math.PR↗

arXiv2018

Explosion and distances in scale-free percolation

We investigate the weighted scale-free percolation (SFPW) model on $\mathbb Z^d$. In the SFPW model, the vertices of $\mathbb Z^d$ are assigned i.i.d. weights $(W_x)_{x\in \mathbb Z^d}$, following a power-law distribution with tail exponent $τ>1$. Conditioned on the collection of weights, the edges $(x,y)_{x, y \in \mathbb Z^d}$ are present independently with probability that a Poisson random variable with parameter $λW_x W_y/(\|x-y\|)^α$ is at least one, for some $α, λ>0$, and where $\|\cdot \|$ denotes the Euclidean distance. After the graph is constructed this way, we assign independent and identically distributed (i.i.d.) random edge lengths from distribution $L$ to all existing edges in the graph. The focus of the paper is to determine when is the obtained model \emph{explosive}, that is, when it is possible to reach infinitely many vertices in finite time from a vertex. We show that explosion happens precisely for those edge length distributions that produce explosive branching processes with infinite mean power law offspring distributions. For non-explosive edge-length distributions, when $γ\in (1,2)$, we characterise the asymptotic behaviour of the time it takes to reach the first vertex that is graph distance $n$ away. For $γ>2$, we show that the number of vertices reachable by time $t$ from the origin grows at most exponentially, thus explosion is never possible. For the non-explosive edge-length distributions, when $γ\in (1,2)$, we further determine the first order asymptotics of distances when $γ\in (1,2)$. As a corollary we obtain a sharp upper and lower bound for graph distances, closing a gap between a lower and upper bound on graph distances in Deijfen Hofstad `13 when $γ\in(1,2), τ>2$.

math.PR↗

arXiv2018

Diameter in ultra-small scale-free random graphs: Extended version

It is well known that many random graphs with infinite variance degrees are ultrasmall. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least $k$ is approximately $k^{-(τ-1)}$ with $τ\in(2,3)$, typical distances between pairs of vertices in a graph of size $n$ are asymptotic to $\frac{2\log\log n}{|\log(τ-2)|}$ and $\frac{4\log\log n}{|\log(τ-2)|}$, respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order $\log\log n$ precisely when the minimal forward degree $d$ of vertices is at least $2$. We identify the exact constant, which equals that of the typical distances plus $2/\log d$. Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.

math.PR↗

arXiv2018

Mixing times of random walks on dynamic configuration models

The mixing time of a random walk, with or without backtracking, on a random graph generated according to the configuration model on $n$ vertices, is known to be of order $\log n$. In this paper we investigate what happens when the random graph becomes {\em dynamic}, namely, at each unit of time a fraction $α_n$ of the edges is randomly rewired. Under mild conditions on the degree sequence, guaranteeing that the graph is locally tree-like, we show that for every $\varepsilon\in(0,1)$ the $\varepsilon$-mixing time of random walk without backtracking grows like $\sqrt{2\log(1/\varepsilon)/\log(1/(1-α_n))}$ as $n \to \infty$, provided that $\lim_{n\to\infty} α_n(\log n)^2=\infty$. The latter condition corresponds to a regime of fast enough graph dynamics. Our proof is based on a randomised stopping time argument, in combination with coupling techniques and combinatorial estimates. The stopping time of interest is the first time that the walk moves along an edge that was rewired before, which turns out to be close to a strong stationary time.

math.PR↗

arXiv2017

Switch chain mixing times through triangle counts

Sampling uniform simple graphs with power-law degree distributions with degree exponent $τ\in(2,3)$ is a non-trivial problem. We propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs. We then compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model. Using simulations and heuristic arguments, we conjecture that the number of triangles in the erased configuration model is larger than the number of triangles in the uniform random graph, provided that the graph is sufficiently large.

math.PR↗

arXiv2017

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.

math.PR↗