arXiv subjects

Remco van der Hofstad

Publications and source records attributed to Remco van der Hofstad.

At least 55 records · Page 3Linked to original sources

arXiv2023

Invasion Percolation on Power-Law Branching Processes

We analyse the cluster discovered by invasion percolation on a branching process with a power-law offspring distribution. Invasion percolation is a paradigm model of self-organised criticality, where criticality is approached without tuning any parameter. By performing invasion percolation for $n$ steps, and letting $n\to\infty$, we find an infinite subtree, called the invasion percolation cluster (IPC). A notable feature of the IPC is its geometry that consists of a unique path to infinity (also called the backbone) onto which finite forests are attached. Our main theorem shows the volume scaling limit of the $k$-cut IPC, which is the cluster containing the root when the edge between the $k$-th and $(k+1)$-st backbone vertices is cut. We assume a power-law offspring distribution with exponent $α$ and analyse the IPC for different power-law regimes. In a finite-variance setting $(α>2)$ our results are a natural extension of previous works on the branching process tree (Michelen et al. 2019) and the regular tree (Angel et al. 2008). However, for an infinite-variance setting ($α\in(1,2)$) or even an infinite-mean setting ($α\in(0,1)$), results significantly change. This is illustrated by the volume scaling of the $k$-cut IPC, which scales as $k^2$ for $α>2$, but as $k^{α/(α-1)}$ for $α\in (1,2)$ and exponentially for $α\in (0,1)$.

math.PR↗

arXiv2023

Detecting a late changepoint in the preferential attachment model

Motivated by the problem of detecting a change in the evolution of a network, we consider the preferential attachment random graph model with a time-dependent attachment function. Our goal is to detect whether the attachment mechanism changed over time, based on a single snapshot of the network and without directly observable information about the dynamics. We cast this question as a hypothesis testing problem, where the null hypothesis is a preferential attachment model with a constant affine attachment parameter $δ_0$, and the alternative hypothesis is a preferential attachment model where the affine attachment parameter changes from $δ_0$ to $δ_1$ at an unknown changepoint time $τ_n$. For our analysis we focus on the regime where $δ_0$ and $δ_1$ are fixed, and the changepoint occurs close to the observation time of the network (i.e., $τ_n = n - c n^γ$ with $c>0$ and $γ\in (0, 1)$). This corresponds to the relevant scenario where we aim to detect the changepoint shortly after it has happened. We present two tests based on the number of vertices with minimal degree, and show that these are asymptotically powerful when $\tfrac{1}{2}<γ<1$. We conjecture that there is no powerful test based on the final network snapshot when $γ< \tfrac{1}{2}$. The first test we propose requires knowledge of $δ_0$. The second test is significantly more involved, and does not require the knowledge of $δ_0$ while still achieving the same performance guarantees. Furthermore, we prove that the test statistics for both tests are asymptotically normal, allowing for accurate calibration of the tests. This is demonstrated by numerical experiments, that also illustrate the finite sample test properties.

math.ST↗

arXiv2023

Dynamic random intersection graph: Dynamic local convergence and giant structure

Random intersection graphs containing an underlying community structure are a popular choice for modelling real-world networks. Given the group memberships, the classical random intersection graph is obtained by connecting individuals when they share at least one group. We extend this approach and make the communities dynamic by letting them alternate between an active and inactive phase. We analyse the new model, delivering results on degree distribution, local convergence, giant component, and maximum group size, paying particular attention to the dynamic description of these properties. We also describe the connection between our model and the bipartite configuration model, which is of independent interest.

math.PR↗

arXiv2023

Universal `winner-takes-it-all' phenomenon in scale-free random graphs

We study competition on scale-free random graphs, where the degree distribution satisfies an asymptotic power-law with infinite variance. Our competition process is such that the two types attempt at occupying vertices incident to the presently occupied sets, and the passage times are independent and identically distributed, possibly with different distributions for the two types. Once vertices are occupied by a type, they remain on being so. We focus on the explosive setting, where our main result shows that the winning type occupies all but a finite number of vertices. This universal `winner-takes-it-all' phenomenon significantly generalises previous work with Deijfen for exponential edge-weights, and considerably simplifies its proof.

math.PR↗

arXiv2023

Scaling of the clustering function in spatial inhomogeneous random graphs

We consider an infinite spatial inhomogeneous random graph model with an integrable connection kernel that interpolates nicely between existing spatial random graph models. Key examples are versions of the weight-dependent random connection model, the infinite geometric inhomogeneous random graph, and the age-based random connection model. These infinite models arise as the local limit of the corresponding finite models, see \cite{LWC_SIRGs_2020}. For these models we identify the scaling of the \emph{local clustering} as a function of the degree of the root in different regimes in a unified way. We show that the scaling exhibits phase transitions as the interpolation parameter moves across different regimes. In addition to the scaling we also identify the leading constants of the clustering function. This allows us to draw conclusions on the geometry of a \emph{typical} triangle contributing to the clustering in the different regimes.

math.PR↗

arXiv2023

It is hard to kill fake news

We study a model for the spread of fake news, where first a piece of fake news is spread from a location in a network, followed by a correction to the news. We assume that both the fake as well as correct news travel as first-passage percolations or SI epidemics with i.i.d.\ traversal times, possibly with different distributions and dependence. We make the (hopeful) assumption that once a vertex in the network hears the correct news, they are immediately convinced that this is indeed correct, and continue to spread the correct news. Even in this optimistic scenario, our main results show that it is very difficult to remove the fake news from the network, even when the correct news would spread much faster than the fake news. We show this on the configuration model, a model that has gained enormous popularity as a simple, yet flexible, model for real-world networks. The crux of the proof is the realization that this problem on a branching process tree (which is known to be the local limit of the configuration model) can be reformulated in terms of the maximum of a branching random walk, a topic that has attracted considerable attention in the past decade. We then extend the results to the graph setting using couplings to branching processes, local convergence and detailed estimates on first-passage percolation on random graphs as derived in \cite{BhaHofHoo17}. Remarkably, despite the fact that our proofs for the configuration model rely on its local branching process structure, the condition for strong survival on a finite number of generations of a branching process tree is {\em different} from that on the configuration model.

math.PR↗

arXiv2023

Connectivity of random graphs after centrality-based vertex removal

Centrality measures aim to indicate who is important in a network. Various notions of `being important' give rise to different centrality measures. In this paper, we study how important the central vertices are for the connectivity structure of the network, by investigating how the removal of the most central vertices affects the number of connected components and the size of the giant component. We use local convergence techniques to identify the limiting number of connected components for locally converging graphs and centrality measures that depend on the vertex's neighborhood. For the size of the giant, we prove a general upper bound. For the matching lower bound, we specialize to the case of degree centrality on one of the most popular models in network science, the configuration model, for which we show that removal of the highest-degree vertices destroys the giant most.

math.PR↗

arXiv2023

The rank of sparse symmetric matrices over arbitrary fields

Let $\FF$ be an arbitrary field and $(\bm{G}_{n,d/n})_n$ be a sequence of sparse weighted Erdős-Rényi random graphs on $n$ vertices with edge probability $d/n$, where weights from $\FF \setminus\{0\}$ are assigned to the edges according to a fixed matrix $J_n$. We show that the normalised rank of the adjacency matrix of $(\bm{G}_{n,d/n})_n$ converges in probability to a constant, and derive the limiting expression. Our result shows that for the general class of sparse symmetric matrices under consideration, the asymptotics of the normalised rank are independent of the edge weights and even the field, in the sense that the limiting constant for the general case coincides with the one previously established for adjacency matrices of sparse (non-weighted) Erdős-Rényi matrices over $\RR$ from \cite{bordenave2011rank}. Our proof, which is purely combinatorial in its nature, is based on an intricate extension of the novel perturbation approach from \cite{coja2022rank} to the symmetric setting.

math.CO↗

arXiv2022

Two more ways of spelling Gini Coefficient with Applications

In this paper, we draw attention to a promising yet slightly underestimated measure of variability - the Gini coefficient. We describe two new ways of defining and interpreting this parameter. Using our new representations, we compute the Gini index for a few probability distributions and describe it in more detail for the negative binomial distribution. We also suggest the latter as a tool to measure overdispersion in epidemiology.

math.ST↗

arXiv2022

Local limits of spatial inhomogeneous random graphs

Consider a set of $n$ vertices, where each vertex has a location in $\mathbb{R}^d$ that is sampled uniformly from the unit cube in $\mathbb{R}^d$, and a weight associated to it. Construct a random graph by placing edges independently for each vertex pair with a probability that is a function of the distance between the locations, and the vertex weights. Under appropriate integrability assumptions on the edge probabilities that imply sparseness of the model, after appropriately blowing up the locations, we prove that the local limit of this random graph sequence is the (countably) infinite random graph on $\mathbb{R}^d$ with vertex locations given by a homogeneous Poisson point process, having weights which are i.i.d. copies of limiting vertex weights. Our setup covers many sparse geometric random graph models from the literature, including Geometric Inhomogeneous Random Graphs (GIRGs), Hyperbolic Random Graphs, Continuum Scale-Free Percolation and Weight-dependent Random Connection Models. We prove that the limiting degree distribution is mixed Poisson, and the typical degree sequence is uniformly integrable, and obtain convergence results on various measures of clustering in our graphs as a consequence of local convergence. Finally, as a by-product of our argument, we prove a doubly logarithmic lower bound on typical distances in this general setting.

math.PR↗

arXiv2022

Rank-k random graphs and finite type branching processes

In this note, we investigate fundamental relations between exploration processes in random graphs, and branching processes. We formulate a class of models that we call {\em rank-$k$ random graphs}, and that are special in that their neighborhood explorations can be obtained by a {\em thinning} of multi-type branching processes. We show that any rank-2 random graph can be described in terms of thinning of a 2-type branching process, while for higher rank, it is not clear how many types are needed.

math.PR↗

arXiv2022

Global lower mass-bound for critical configuration models in the heavy-tailed regime

We establish the global lower mass-bound property for the largest connected components in the critical window for the configuration model when the degree distribution has an infinite third moment. The scaling limit of the critical percolation clusters, viewed as measured metric spaces, was established in [7] with respect to the Gromov-weak topology. Our result extends those scaling limit results to the stronger Gromov-Hausdorff-Prokhorov topology under slightly stronger assumptions on the degree distribution. This implies the distributional convergence of global functionals such as the diameter of the largest critical components. Further, our result gives a sufficient condition for compactness of the random metric spaces that arise as scaling limits of critical clusters in the heavy-tailed regime.

math.PR↗

arXiv2022

Critical window for connectivity in the Configuration Model

We identify the asymptotic probability of a configuration model $\mathrm{CM}_n(\boldsymbol{d})$ to produce a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which the asymptotic number of simple connected graphs with a prescribed degree sequence follows.

math.PR↗

arXiv2022

The winner takes it all but one

We study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with type 1 and type 2 infection, respectively, and the infection then spreads via nearest neighbors in the graph. The time it takes for the type 1 (resp. 2) infection to traverse an edge $e$ is given by a random variable $X_1(e)$ (resp. $X_2(e)$) and, if the vertex at the other end of the edge is still uninfected, it then becomes type 1 (resp. 2) infected and immune to the other type. Assuming that the degrees follow a power-law distribution with exponent $τ\in (1,2)$, we show that, with high probability as the number of vertices tends to infinity, one of the infection types occupies all vertices except for the starting point of the other type. Moreover, both infections have a positive probability of winning regardless of the passage times distribution. The result is also shown to hold for the erased configuration model, where self-loops are erased and multiple edges are merged, and when the degrees are conditioned to be smaller than $n^α$ for some $α> 0$.

math.PR↗

arXiv2022

The role of inter-regional mobility in forecasting SARS-CoV-2 transmission

In this paper, we present a method to forecast the spread of SARS-CoV-2 across regions with a focus on the role of mobility. Mobility has previously been shown to play a significant role in the spread of the virus, particularly between regions. Here, we investigate under which epidemiological circumstances incorporating mobility into transmission models yields improvements in the accuracy of forecasting, where we take the situation in the Netherlands during and after the first wave of transmission in 2020 as a case study. We assess the quality of forecasting on the detailed level of municipalities, instead of on a nation-wide level. To model transmissions, we use a simple mobility-enhanced SEIR compartmental model with subpopulations corresponding to the Dutch municipalities. We use commuter information to quantify mobility, and develop a method based on maximum likelihood estimation to determine the other relevant parameters. We show that taking inter-regional mobility into account generally leads to an improvement in forecast quality. However, at times when policies are in place that aim to reduce contacts or travel, this improvement is very small.

math.OC↗

arXiv2022

The Hyperspherical Geometry of Community Detection: Modularity as a Distance

We introduce a metric space of clusterings, where clusterings are described by a binary vector indexed by the vertex-pairs. We extend this geometry to a hypersphere and prove that maximizing modularity is equivalent to minimizing the angular distance to some modularity vector over the set of clustering vectors. In that sense, modularity-based community detection methods can be seen as a subclass of a more general class of projection methods, which we define as the community detection methods that adhere to the following two-step procedure: first, mapping the network to a point on the hypersphere; second, projecting this point to the set of clustering vectors. We show that this class of projection methods contains many interesting community detection methods. Many of these new methods cannot be described in terms of null models and resolution parameters, as is customary for modularity-based methods. We provide a new characterization of such methods in terms of meridians and latitudes of the hypersphere. In addition, by relating the modularity resolution parameter to the latitude of the corresponding modularity vector, we obtain a new interpretation of the resolution limit that modularity maximization is known to suffer from.

cs.SI↗

arXiv2022

The number of partial Steiner systems and $d$-partitions

We prove asymptotic upper bounds on the number of $d$-partitions (paving matroids of fixed rank) and partial Steiner systems (sparse paving matroids of fixed rank), using a mixture of entropy counting, sparse encoding, and the probabilistic method.

math.CO↗

arXiv2021

Annealed inhomogeneities in random ferromagnets

We consider spin models on complex networks frequently used to model social and technological systems. We study the annealed ferromagnetic Ising model for random networks with either independent edges (Erdős-Rényi), or with prescribed degree distributions (configuration model). Contrary to many physical models, the annealed setting is poorly understood and behaves quite differently than the quenched system. In annealed networks with a fluctuating number of edges, the Ising model changes the degree distribution, an aspect previously ignored. For random networks with Poissonian degrees, this gives rise to three distinct annealed critical temperatures depending on the precise model choice, only one of which reproduces the quenched one. In particular, two of these annealed critical temperatures are finite even when the quenched one is infinite, since then the annealed graph creates a giant component for all sufficiently small temperatures. We see that the critical exponents in the configuration model with deterministic degrees are the same as the quenched ones, which are the mean-field exponents if the degree distribution has finite fourth moment, and power-law-dependent critical exponents otherwise. Remarkably, the annealing for the configuration model with random i.i.d. degrees washes away the universality class with power-law critical exponents.

cond-mat.dis-nn↗