arXiv subjects

Remco van der Hofstad

Publications and source records attributed to Remco van der Hofstad.

At least 37 records · Page 2Linked to original sources

arXiv2025

Local limit of Prim's algorithm

We study the local evolution of Prim's algorithm on large finite weighted graphs. When performed for $n$ steps, where $n$ is the size of the graph, Prim's algorithm will construct the minimal spanning tree (MST). We assume that our graphs converge locally in probability to some limiting rooted graph. In that case, Aldous and Steele already proved that the local limit of the MST converges to a limiting object, which can be thought of as the MST on the limiting infinite rooted graph. Our aim is to investigate {\em how} the local limit of the MST is reached \textit{dynamically}. For this, we take $tn+o(n)$ steps of Prim, for $t\in[0,1]$, and, under some reasonable assumptions, show how the local structure interpolates between performing Prim's algorithm on the local limit when $t=0$, to the full local limit of the MST for $t=1$. Our proof relies on the use of the recently developed theory of {\em dynamic local convergence}. We further present several examples for which our assumptions, and thus our results, apply.

math.PR↗

arXiv2025

Degrees in Preferential Attachment Networks with an Anomaly

We consider a preferential attachment model that incorporates an anomaly. Our goal is to understand the evolution of the network before and after the occurrence of the anomaly by studying the influence of the anomaly on the structural properties of the network. The anomaly is such that after its arrival it attracts newly added edges with fixed probability. We investigate the growth of degrees in the network, finding that the anomaly's degree increases almost linearly. We also provide a heuristic derivation for the exponent of the limiting degree distributions of ordinary vertices, and study the degree growth of the oldest vertex. We show that when the anomaly enters early, the degree distribution is altered significantly, while a late anomaly has minimal impact. Our analysis provides deeper insights into the evolution of preferential attachment networks with an anomalous vertex.

physics.soc-ph↗

arXiv2025

Ising model on preferential attachment models

We study the Ising model on affine preferential attachment models with general parameters. We identify the thermodynamic limit of several quantities, arising in the large graph limit, such as pressure per particle, magnetisation, and internal energy for these models. Furthermore, for $m\geq 2$, we determine the inverse critical temperature for preferential attachment models as $β_c(m,δ)=0$ when $δ\in(-m,0]$, while, for $δ>0$, $$β_c(m,δ)= {\rm atanh}\left\{ \fracδ{2\big( m(m+δ)+\sqrt{m(m-1)(m+δ)(m+δ+1)} \big)} \right\}~.$$ Our proof for the thermodynamic limit of pressure per particle critically relies on the belief propagation theory for factor models on locally tree-like graphs, as developed by Dembo, Montanari, and Sun. It has been proved that preferential attachment models admit the Pólya point tree as their local limit under general conditions. We use the explicit characterisation of the Pólya point tree and belief propagation for factor models to obtain the explicit expression for the thermodynamic limit of the pressure per particle. Next, we use the convexity properties of the internal energy and magnetisation to determine their thermodynamic limits. To study the phase transition, we prove that the inverse critical temperature for a sequence of graphs and its local limit are equal. Finally, we show that $β_c(m,δ)$ is the inverse critical temperature for the Pólya point tree with parameters $m$ and $δ$, using results from Lyons who shows that the critical inverse temperature is closely related to the percolation critical threshold. This part of the proof heavily relies on the critical percolation threshold for Pólya point trees established earlier with Hazra.

math.PR↗

arXiv2025

Value of risk-contact data from digital contact monitoring apps in infectious disease modeling

In this paper, we present a simple method to integrate risk-contact data, obtained via digital contact monitoring (DCM) apps, in conventional compartmental transmission models. During the recent COVID-19 pandemic, many such data have been collected for the first time via newly developed DCM apps. However, it is unclear what the added value of these data is, unlike that of traditionally collected data via, e.g., surveys during non-epidemic times. The core idea behind our method is to express the number of infectious individuals as a function of the proportion of contacts that were with infected individuals and use this number as a starting point to initialize the remaining compartments of the model. As an important consequence, using our method, we can estimate key indicators such as the effective reproduction number using only two types of daily aggregated contact information, namely the average number of contacts and the average number of those contacts that were with an infected individual. We apply our method to the recent COVID-19 epidemic in the Netherlands, using self-reported data from the health surveillance app COVID RADAR and proximity-based data from the contact tracing app CoronaMelder. For both data sources, our corresponding estimates of the effective reproduction number agree both in time and magnitude with estimates based on other more detailed data sources such as daily numbers of cases and hospitalizations. This suggests that the use of DCM data in transmission models, regardless of the precise data type and for example via our method, offers a promising alternative for estimating the state of an epidemic, especially when more detailed data are not available.

q-bio.PE↗

arXiv2025

Random cluster models on random graphs

On locally tree-like random graphs, we relate the random cluster model with external magnetic fields and $q\geq 2$ to Ising models with vertex-dependent external fields. The fact that one can formulate general random cluster models in terms of two-spin ferromagnetic Ising models is quite interesting in its own right. However, in the general setting, the external fields are both positive and negative, which is mathematically unexplored territory. Interestingly, due to the reformulation as a two-spin model, we can show that the Bethe partition function, which is believed to have the same pressure per particle, is always a {\em lower bound} on the graph pressure per particle. We further investigate special cases in which the external fields do always have the same sign. The first example is the Potts model with general external fields on random $d$-regular graphs. In this case, we show that the pressure per particle in the quenched setting agrees with that of the annealed setting, and verify \cite[Assumption 1.4]{BasDemSly23}. We show that there is a line of values for the external fields where the model displays a first-order phase transition. This completes the identification of the phase diagram of the Potts model on the random $d$-regular graph. As a second example, we consider the high external field and low temperature phases of the system on locally tree-like graphs with general degree distribution.

math.PR↗

arXiv2025

Annealed Potts models on rank-1 inhomogeneous random graphs

In this paper, we study the annealed ferromagnetic $q$-state Potts model on sparse rank-1 random graphs, where vertices are equipped with a vertex weight, and the probability of an edge is proportional to the product of the vertex weights. In an annealed system, we take the average on both numerator and denominator of the ratio defining the Boltzmann-Gibbs measure of the Potts model. We show that the thermodynamic limit of the pressure per particle exists for rather general vertex weights. In the infinite-variance weight case, we show that the critical temperature equals infinity. For finite-variance weights, we show that, under a rather general condition, the phase transition is {\em first order} for all $q\geq 3$. However, we cannot generally show that the discontinuity of the order parameter is {\em unique}. We prove this uniqueness under a reasonable condition that holds for various distributions, including uniform, gamma, log-normal, Rayleigh and Pareto distributions. Further, we show that the first-order phase transition {\em persists} even for some small positive external field. In the rather relevant case of Pareto distributions with power-law exponent $τ$, remarkably, the phase transition is first order when $τ\geq 4$, but not necessarily when the weights have an infinite third-moment, i.e., when $τ\in(3,4)$. More precisely, the phase transition is second order for $τ\in (3,τ(q)]$, while it is first order when $τ>τ(q)$, where we give an explicit equation that $τ(q)$ solves.

math.PR↗

arXiv2025

SIR on locally converging dynamic random graphs

In this paper, we study the trajectory of a classic SIR epidemic on a family of dynamic random graphs of fixed size, whose set of edges continuously evolves over time. We set general infection and recovery times, and start the epidemic from a positive, yet small, proportion of vertices. We show that in such a case, the spread of an infectious disease around a typical individual can be approximated by the spread of the disease in a local neighbourhood of a uniformly chosen vertex. We formalize this by studying general dynamic random graphs that converge dynamically locally in probability and demonstrate that the epidemic on these graphs converges to the epidemic on their dynamic local limit graphs. We provide a detailed treatment of the theory of dynamic local convergence, which remains a relatively new topic in the study of random graphs. One main conclusion of our paper is that a specific form of dynamic local convergence is required for our results to hold.

math.PR↗

arXiv2024

Power-law hypothesis for PageRank on undirected graphs

Based on observations in the web-graph, the power-law hypothesis states that PageRank has a power-law distribution with the same exponent as the in-degree. While this hypothesis has been analytically verified for many random graph models, such as directed configuration models and generalized random graphs, surprisingly it has recently been disproven for the directed preferential attachment model. In this paper, we prove that in undirected networks, the graph-normalized PageRank is always upper bounded by the degree. Furthermore, we prove that the corresponding (asymptotic) lower bound holds true under reasonable assumptions on the local weak limit, but not in general, and we provide a counterexample. Our result shows that PageRank always has a lighter tail than the degree, which contrasts the case of the directed preferential attachment model, where PageRank has a heavier tail instead. We end the paper with a discussion, where we extend our results to directed networks with a bounded ratio of in- and out-degrees, and reflect on our methods by contrasting the undirected and directed preferential attachment model.

math.PR↗

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↗

arXiv2024

Learning the mechanisms of network growth

We propose a novel model-selection method for dynamic networks. Our approach involves training a classifier on a large body of synthetic network data. The data is generated by simulating nine state-of-the-art random graph models for dynamic networks, with parameter range chosen to ensure exponential growth of the network size in time. We design a conceptually novel type of dynamic features that count new links received by a group of vertices in a particular time interval. The proposed features are easy to compute, analytically tractable, and interpretable. Our approach achieves a near-perfect classification of synthetic networks, exceeding the state-of-the-art by a large margin. Applying our classification method to real-world citation networks gives credibility to the claims in the literature that models with preferential attachment, fitness and aging fit real-world citation networks best, although sometimes, the predicted model does not involve vertex fitness.

cs.SI↗

arXiv2024

Euclidean TSP in Narrow Strips

We investigate how the complexity of Euclidean TSP for point sets $P$ inside the strip $(-\infty,+\infty)\times [0,δ]$ depends on the strip width $δ$. We obtain two main results. First, for the case where the points have distinct integer $x$-coordinates, we prove that a shortest bitonic tour (which can be computed in $O(n\log^2 n)$ time using an existing algorithm) is guaranteed to be a shortest tour overall when $δ\leq 2\sqrt{2}$, a bound which is best possible. Second, we present an algorithm that is fixed-parameter tractable with respect to $δ$. Our algorithm has running time $2^{O(\sqrtδ)} n + O(δ^2 n^2)$ for sparse point sets, where each $1\timesδ$ rectangle inside the strip contains $O(1)$ points. For random point sets, where the points are chosen uniformly at random from the rectangle $[0,n]\times [0,δ]$, it has an expected running time of $2^{O(\sqrtδ)} n$. These results generalise to point sets $P$ inside a hypercylinder of width $δ$. In this case, the factors $2^{O(\sqrtδ)}$ become $2^{O(δ^{1-1/d})}$.

cs.CG↗

arXiv2024

Are giants in random digraphs `almost' local?

Recently, the first author showed that the giant in random undirected graphs is `almost' local. This means that, under a necessary and sufficient condition, the limiting proportion of vertices in the giant converges in probability to the survival probability of the local limit. We extend this result to the setting of random digraphs, where connectivity patterns are significantly more subtle. For this, we identify the precise version of local convergence for digraphs that is needed. We also determine bounds on the number of strongly connected components, and calculate its asymptotics explicitly for locally tree-like digraphs, as well as for other locally converging digraph sequences under the `almost-local' condition for the strong giant. The fact that the number of strongly connected components is {\em not} local once more exemplifies the delicate nature of strong connectivity in random digraphs.

math.PR↗

arXiv2024

Mixing of fast random walks on dynamic random permutations

We analyse the mixing profile of a random walk on a dynamic random permutation, focusing on the regime where the walk evolves much faster than the permutation. Two types of dynamics generated by random transpositions are considered: one allows for coagulation of permutation cycles only, the other allows for both coagulation and fragmentation. We show that for both types, after scaling time by the length of the permutation and letting this length tend to infinity, the total variation distance between the current distribution and the uniform distribution converges to a limit process that drops down in a single jump. This jump is similar to a one-sided cut-off, occurs after a random time whose law we identify, and goes from the value 1 to a value that is a strictly decreasing and deterministic function of the time of the jump, related to the size of the largest component in Erdős-Rényi random graphs. After the jump, the total variation distance follows this function down to 0.

math.PR↗

arXiv2023

Communication protocol for a satellite-swarm interferometer

Orbiting low frequency antennas for radio astronomy (OLFAR) that capture cosmic signals in the frequency range below 30MHz could provide valuable insights on our Universe. These wireless swarms of satellites form a connectivity graph that allows data exchange between most pairs of satellites. Since this swarm acts as an interferometer, the aim is to compute the cross-correlations between most pairs of satellites. We propose a k-nearest-neighbour communication protocol, and investigate the minimum neighbourhood size of each satellite that ensures connectivity of at least 95% of the swarm. We describe the proportion of cross-correlations that can be computed in our method given an energy budget per satellite. Despite the method's apparent simplicity, it allows us to gain insight into the requirements for such satellite swarms. In particular, we give specific advice on the energy requirements to have sufficient coverage of the relevant baselines.

eess.SP↗

arXiv2023

The Projection Method: a Unified Formalism for Community Detection

We present the class of projection methods for community detection that generalizes many popular community detection methods. In this framework, we represent each clustering (partition) by a vector on a high-dimensional hypersphere. A community detection method is a projection method if it can be described by the following two-step approach: 1) the graph is mapped to a query vector on the hypersphere; and 2) the query vector is projected on the set of clustering vectors. This last projection step is performed by minimizing the distance between the query vector and the clustering vector, over the set of clusterings. We prove that optimizing Markov stability, modularity, the likelihood of planted partition models and correlation clustering fit this framework. A consequence of this equivalence is that algorithms for each of these methods can be modified to perform the projection step in our framework. In addition, we show that these different methods suffer from the same granularity problem: they have parameters that control the granularity of the resulting clustering, but choosing these to obtain clusterings of the desired granularity is nontrivial. We provide a general heuristic to address this granularity problem, which can be applied to any projection method. Finally, we show how, given a generator of graphs with community structure, we can optimize a projection method for this generator in order to obtain a community detection method that performs well on this generator.

cs.SI↗

arXiv2023

On a Near-Optimal \& Efficient Algorithm for the Sparse Pooled Data Problem

The pooled data problem asks to identify the unknown labels of a set of items from condensed measurements. More precisely, given $n$ items, assume that each item has a label in $\cbc{0,1,\ldots, d}$, encoded via the ground-truth $\SIGMA$. We call the pooled data problem sparse if the number of non-zero entries of $\SIGMA$ scales as $k \sim n^θ$ for $θ\in (0,1)$. The information that is revealed about $\SIGMA$ comes from pooled measurements, each indicating how many items of each label are contained in the pool. The most basic question is to design a pooling scheme that uses as few pools as possible, while reconstructing $\SIGMA$ with high probability. Variants of the problem and its combinatorial ramifications have been studied for at least 35 years. However, the study of the modern question of \emph{efficient} inference of the labels has suggested a statistical-to-computational gap of order $\log n$ in the minimum number of pools needed for theoretically possible versus efficient inference. In this article, we resolve the question whether this $\log n$-gap is artificial or of a fundamental nature by the design of an efficient algorithm, called \algoname, based upon a novel pooling scheme on a number of pools very close to the information-theoretic threshold.

math.PR↗

arXiv2023

Percolation on preferential attachment models

We study the percolation phase transition on preferential attachment models, in which vertices enter with $m$ edges and attach proportionally to their degree plus $δ$. We identify the critical percolation threshold as $$π_c=\fracδ{2\big(m(m+δ)+\sqrt{m(m-1)(m+δ)(m+1+δ)}\big)}$$ for $δ$ positive and $π_c=0$ for non-positive values of $δ$. Therefore the giant component is robust for $δ\in(-m,0]$, while it is not for $δ>0$. Our proof for the critical percolation threshold consists of three main steps. First, we show that preferential attachment graphs are large-set expanders, enabling us to verify the conditions outlined by Alimohammadi, Borgs, and Saberi (2023). Within their conditions, the proportion of vertices in the largest connected component in a sequence converges to the survival probability of percolation on the local limit. In particular, the critical percolation threshold for both the graph and its local limit are identical. Second, we identify $1/π_c$ as the spectral radius of the mean offspring operator of the Pólya point tree, the local limit of preferential attachment models. Lastly, we prove that the critical percolation threshold for the Pólya point tree is the inverse of the spectral radius of the mean offspring operator. For positive $δ$, we use sub-martingales to prove sub-criticality and apply spine decomposition theory to demonstrate super-criticality, completing the third step of the proof. For $δ\leq 0$ and any $π>0$ instead, we prove that the percolated Pólya point tree dominates a supercritical branching process, proving that the critical percolation threshold equals $0$.

math.PR↗

arXiv2023

Lace Expansion and Mean-Field Behavior for the Random Connection Model

We study the random connection model driven by a stationary Poisson process. In the first part of the paper, we derive a lace expansion with remainder term in the continuum and bound the coefficients using a new version of the BK inequality. For our main results, we consider three versions of the connection function $φ$: a finite-variance version (including the Boolean model), a spread-out version, and a long-range version. For sufficiently large dimension (resp., spread-out parameter and $d>6$), we then prove the convergence of the lace expansion, derive the triangle condition, and establish an infra-red bound. From this, mean-field behavior of the model can be deduced. As an example, we show that the critical exponent $γ$ takes its mean-field value $γ=1$ and that the percolation function is continuous.

math.PR↗