arXiv subjects

Pim van der Hoorn

Publications and source records attributed to Pim van der Hoorn.

At least 19 recordsLinked to original sources

arXiv2026

Limiting Behavior of Degree-Degree Metrics under Local Convergence in Probability

This paper investigates the limiting behaviour of degree-degree correlation metrics for sequences of random graphs under a general assumption of local convergence in probability. We establish convergence results for Pearson's correlation coefficient r, Spearman's rho, Kendall's tau, average nearest neighbour degree (ANND), and average nearest neighbour rank (ANNR). Our results explicitly show how the limits of these degree-degree correlation metrics depend on the local structure of the graph. We then apply our general results to study degree-degree correlations in rank-1 inhomogeneous random graphs and random geometric graphs, deriving explicit expressions for ANND in both models and for Pearson's correlation coefficient in the latter one. Keywords: random graphs, degree-degree metrics, neutral mixing, geometric random graphs

math.PR↗

arXiv2026

Flexible random graph modeling for cell-cell interactions

In this paper we explore generalisations to a random digraph model designed for modelling cell-cell interactions (RaCInG). In its baseline model -- tested both theoretically and in biological practice -- a fixed number of arcs are assigned uniformly at random to vertices that have suitable types to accept them. Generalisations to this model are explored in two directions: creating undirected graphs in favour of directed graphs, and incorporating spatial information. Both directions are driven by biological demands: better interpretability and adaptation to technological advances. We show not only that these generalisations are theoretically possible within the framework of RaCInG, but that they can be made rigorous, proving that RaCInG creates a flexible random graph framework to model cell-cell interactions.

math.PR↗

arXiv2026

Projective limits of probabilistic symmetries and their applications to random graph limits

We couple projective limits of probability measures to direct limits of their symmetry groups. We show that the direct limit group is the group of symmetries of the projective limit probability measure. If projective systems of probability measures represent point processes in increasingly larger finite regions of the same infinite space, then we show that under some additional niceness and consistency assumptions, an extension of the direct limit group is the symmetry group of the projective limit point process in the whole infinite space. The application of these results to random graph limits provides ``shortest paths'' to graphons and graphexes as it recovers these random graph limits as trivial corollaries. Another application example encompasses a broad class of limits of random graphs with bounded average degrees. This class includes a representative collection of paradigmatic random graph models that have attracted significant research attention in diverse areas of science. Our approach thus provides a general unified framework to study limits of very different types of random graphs.

math.PR↗

arXiv2025

A Graph Random Walk Method for Calculating Time-of-Flight Charge Mobility in Organic Semiconductors from Multiscale Simulations

We present a graph random walk (GRW) method for the study of charge transport properties of complex molecular materials in the time-of-flight regime. The molecules forming the material are represented by the vertices of a directed weighted graph, and the charge carriers are random walkers. The edge weights are rates for elementary jumping processes for a charge carrier to move along the edge and are determined from a combination of the energies of the involved vertices and an interaction strength. Exclusions are built into the random walk to account for the Pauli exclusion principle. In time-of-flight experiments, charge carriers are injected into the material and the time until they reach a collecting electrode is recorded. Our approach allows direct evaluation of the expected hitting time of the collecting nodes in terms of a sparse, linear system, avoiding numerically cumbersome and potentially fluctuations-prone methods based on explicit time evolution from solutions of a high-dimensional Master Equation or from kinetic Monte Carlo (KMC). We validate the GRW approach by numerical studies of charge dynamics of single and multiple carriers in diffusive and drift-diffusive regimes using a surrogate lattice model of a realistic material whose properties have been simulated within a multiscale model framework combining quantum-mechanical and molecular-mechanics methods. The surrogate model allows varying types and strengths of energetic disorder from the reference baseline. Comparison with results from the Master Equation confirms the theoretical equivalence of both approaches also in numerical implementations. We further show that KMC results show substantial deviations due to inadequate sampling. All in all, we find that the GRW method provides a powerful alternative to the more commonly used methods without sampling issues and with the benefit of making use of sparse matrix methods.

cond-mat.stat-mech↗

arXiv2025

Automatic Identification of Traps in Molecular Charge Transport Networks of Organic Semiconductors

This paper introduces a method to identify traps in molecular charge transport networks as obtained by multiscale modeling of organic semiconductors. Depending on the materials, traps can be defect-like single molecules or clusters of several neighboring ones, and can have a significant impact on the dynamics of charge carriers. Our proposed method builds on the random walk model of charge dynamics on a directed, weighted graph, the molecular transport network. It comprises an effective heuristic to determine the number of traps or trap clusters based on the eigenvalues and eigenvectors of the random walk Laplacian matrix and uses subsequent spectral clustering techniques to identify these traps. In contrast to currently available methods, ours enables identification of trap molecules in organic semiconductors without having to explicitly simulate the charge dynamics and is applicable to a variety of energy- or topology-based traps in homomolecular or mixed systems with or without detailed-balance. As a prototypical system we simulate an amorphous morphology of bathocuproine, a material with known high energetic disorder and charge trapping. Based on a first-principle multiscale model, we first obtain a reference charge transport network and then purposefully modify its properties to represent different trap characteristics. In contrast to currently available methods, our approach successfully identifies both single trap, multiple distributed traps, and a combination of a single-molecule trap and trap regions on an equal footing.

physics.comp-ph↗

arXiv2025

Uncertainty Quantification in Multiscale Models of Charge Transport in Organic Semiconductors: Influence of the Exhange-Correlation Functional

This study investigates the impact of exchange-correlation functional choices on the predictive accuracy of multiscale models for charge transport in organic semiconductors (OSCs). A hybrid functional approach is applied to analyze uncertainties in key parameters influencing charge mobility, focusing on the Hartree--Fock exchange fraction. Using 2-methyl-9,10-bis(naphthalen-2-yl)anthracene (MADN) as a test system, molecular dynamics and density functional theory are combined to compute reorganization energies, site energies, and electronic coupling elements. Monte Carlo sampling quantifies the uncertainty propagation, revealing that site energy variations dominate transport property uncertainties, while coupling elements contribute minimally. The findings underscore the need for accurate parameter determination and functional selection, with implications for enhancing the reliability of first-principles-based multiscale modeling frameworks in OSC design.

cond-mat.mtrl-sci↗

arXiv2024

Condensation in scale-free geometric graphs with excess edges

We identify the upper large deviation probability for the number of edges in scale-free geometric random graph models as the space volume goes to infinity. Our result covers the models of scale-free percolation, the Boolean model with heavy-tailed radius distribution, and the age-dependent random connection model. In all these cases the mechanism behind the large deviation is based on a condensation effect. Loosely speaking, the mechanism randomly selects a finite number of vertices and increases their power, so that they connect to a macroscopic number of vertices in the graph, while the other vertices retain a degree close to their expectation and thus make no more than the expected contribution to the large deviation event. We verify this intuition by means of limit theorems for the empirical distributions of degrees and edge-lengths under the conditioning. We observe that at large finite volumes, the edge-length distribution splits into a bulk and travelling wave part of asymptotically positive proportions.

math.PR↗

arXiv2023

From inhomogeneous random digraphs to random graphs with fixed arc counts

Consider a random graph model with $n$ vertices where each vertex has a vertex-type drawn from some discrete distribution. Suppose that the number of arcs to be placed between each pair of vertex-types is known, and that each arc is placed uniformly at random without replacement between one of the vertex-pairs with matching types. In this paper, we will show that under certain conditions this random graph model is equivalent to the well-studied inhomogeneous random digraph model. We will use this equivalence in three applications. First, we will apply the equivalence on some well known random graph models (the Erdős-Rényi model, the stochastic block model, and the Chung-Lu model) to showcase what their equivalent counterparts with fixed arcs look like. Secondly, we will extend this equivalence to a practical model for inferring cell-cell interactions to showcase how theoretical knowledge about inhomogeneous random digraphs can be transferred to a modeling context. Thirdly, we will show how our model induces a natural fast algorithm to generate inhomogeneous random digraphs.

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↗

arXiv2022

Entropy of labeled versus unlabeled networks

The structure of a network is an unlabeled graph, yet graphs in most models of complex networks are labeled by meaningless random integers. Is the associated labeling noise always negligible, or can it overpower the network-structural signal? To address this question, we introduce and consider the sparse unlabeled versions of popular network models, and compare their entropy against the original labeled versions. We show that labeled and unlabeled Erdos-Renyi graphs are entropically equivalent, even though their degree distributions are very different. The labeled and unlabeled versions of the configuration model may have different prefactors in their leading entropy terms, although this remains conjectural. Our main results are upper and lower bounds for the entropy of labeled and unlabeled one-dimensional random geometric graphs. We show that their unlabeled entropy is negligible in comparison with the labeled entropy. This means that in sparse networks the entropy of meaningless labeling may dominate the entropy of the network structure. The main implication of this result is that the common practice of using exchangeable models to reason about real-world networks with distinguishable nodes may introduce uncontrolled aberrations into conclusions made about these networks, suggesting a need for a thorough reexamination of the statistical foundations and key results of network science.

physics.soc-ph↗

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↗

arXiv2021

Regular graphs with linearly many triangles

A $d$-regular graph on $n$ nodes has at most $T_{\max} = \frac{n}{3} \tbinom{d}{2}$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular graph has at least $c \cdot T_{\max}$ triangles, and provide a strong structural description of such graphs. When $d$ is fixed, we show that such graphs typically consist of many disjoint $d+1$-cliques and an almost triangle-free part. When $d$ is allowed to grow with $n$, we show that such graphs typically consist of $d+o(d)$ sized almost cliques together with an almost triangle-free part. This confirms a conjecture of Collet and Eckmann from 2002 and considerably strengthens their observation that the triangles cannot be totally scattered in typical instances of regular graphs with many triangles.

math.CO↗

arXiv2021

Ollivier-Ricci curvature convergence in random geometric graphs

Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional definition of curvature of a Riemannian manifold. Here we show that Ollivier curvature of random geometric graphs in any Riemannian manifold converges in the continuum limit to Ricci curvature of the underlying manifold, but only if the definition of Ollivier graph curvature is properly generalized to apply to mesoscopic graph neighborhoods. This result establishes the first rigorous link between a definition of curvature applicable to networks and a traditional definition of curvature of smooth spaces.

math.PR↗

arXiv2020

Clustering in a hyperbolic model of complex networks

In this paper we consider the clustering coefficient and clustering function in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been shown that this model has various properties associated with complex networks, e.g. power-law degree distribution, short distances and non-vanishing clustering coefficient. Here we show that the clustering coefficient tends in probability to a constant $γ$ that we give explicitly as a closed form expression in terms of $α, ν$ and certain special functions. This improves earlier work by Gugelmann et al., who proved that the clustering coefficient remains bounded away from zero with high probability, but left open the issue of convergence to a limiting constant. Similarly, we are able to show that $c(k)$, the average clustering coefficient over all vertices of degree exactly $k$, tends in probability to a limit $γ(k)$ which we give explicitly as a closed form expression in terms of $α, ν$ and certain special functions. We are able to extend this last result also to sequences $(k_n)_n$ where $k_n$ grows as a function of $n$. Our results show that $γ(k)$ scales differently, as $k$ grows, for different ranges of $α$. More precisely, there exists constants $c_{α,ν}$ depending on $α$ and $ν$, such that as $k \to \infty$, $γ(k) \sim c_{α,ν} \cdot k^{2 - 4α}$ if $\frac{1}{2} < α< \frac{3}{4}$, $γ(k) \sim c_{α,ν} \cdot \log(k) \cdot k^{-1} $ if $α=\frac{3}{4}$ and $γ(k) \sim c_{α,ν} \cdot k^{-1}$ when $α> \frac{3}{4}$. These results contradict a claim of Krioukov et al., which stated that the limiting values $γ(k)$ should always scale with $k^{-1}$ as we let $k$ grow.

math.PR↗

arXiv2020

Weighted hypersoft configuration model

Maximum entropy null models of networks come in different flavors that depend on the type of constraints under which entropy is maximized. If the constraints are on degree sequences or distributions, we are dealing with configuration models. If the degree sequence is constrained exactly, the corresponding microcanonical ensemble of random graphs with a given degree sequence is the configuration model per se. If the degree sequence is constrained only on average, the corresponding grand-canonical ensemble of random graphs with a given expected degree sequence is the soft configuration model. If the degree sequence is not fixed at all but randomly drawn from a fixed distribution, the corresponding hypercanonical ensemble of random graphs with a given degree distribution is the hypersoft configuration model, a more adequate description of dynamic real-world networks in which degree sequences are never fixed but degree distributions often stay stable. Here, we introduce the hypersoft configuration model of weighted networks. The main contribution is a particular version of the model with power-law degree and strength distributions, and superlinear scaling of strengths with degrees, mimicking the properties of some real-world networks. As a byproduct, we generalize the notions of sparse graphons and their entropy to weighted networks.

physics.soc-ph↗

arXiv2020

Ollivier curvature of random geometric graphs converges to Ricci curvature of their Riemannian manifolds

Curvature is a fundamental geometric characteristic of smooth spaces. In recent years different notions of curvature have been developed for combinatorial discrete objects such as graphs. However, the connections between such discrete notions of curvature and their smooth counterparts remain lurking and moot. In particular, it is not rigorously known if any notion of graph curvature converges to any traditional notion of curvature of smooth space. Here we prove that in proper settings the Ollivier-Ricci curvature of random geometric graphs in Riemannian manifolds converges to the Ricci curvature of the manifold. This is the first rigorous result linking curvature of random graphs to curvature of smooth spaces. Our results hold for different notions of graph distances, including the rescaled shortest path distance, and for different graph densities. With the scaling of the average degree, as a function of the graph size, ranging from nearly logarithmic to nearly linear.

math.PR↗

arXiv2020

Problems with classification, hypothesis testing, and estimator convergence in the analysis of degree distributions in networks

In their recent work "Scale-free networks are rare", Broido and Clauset address the problem of the analysis of degree distributions in networks to classify them as scale-free at different strengths of "scale-freeness." Over the last two decades, a multitude of papers in network science have reported that the degree distributions in many real-world networks follow power laws. Such networks were then referred to as scale-free. However, due to a lack of a precise definition, the term has evolved to mean a range of different things, leading to confusion and contradictory claims regarding scale-freeness of a given network. Recognizing this problem, the authors of "Scale-free networks are rare" try to fix it. They attempt to develop a versatile and statistically principled approach to remove this scale-free ambiguity accumulated in network science literature. Although their paper presents a fair attempt to address this fundamental problem, we must bring attention to some important issues in it.

physics.soc-ph↗

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↗