arXiv subjects

Frank den Hollander

Publications and source records attributed to Frank den Hollander.

At least 19 recordsLinked to original sources

arXiv2026

The friendship paradox for trees

We analyse the friendship paradox on finite and infinite trees. In particular, we monitor the vertices for which the friendship-bias is positive, neutral and negative, respectively. For an arbitrary finite tree, we show that the number of positive vertices is at least as large as the number of negative vertices, a property we refer to as significance, and derive a lower bound in terms of the branching points in the tree. For an infinite Galton-Watson tree, we compute the densities of the positive and the negative vertices and show that either may dominate the other, depending on the offspring distribution. We also compute the densities of the edges having two given types of vertices at their ends, and give conditions in terms of the offspring distribution under which these types are positively or negatively correlated.

math.PR↗

arXiv2026

Renormalisation of Inhomogeneous Random Graphs

We consider inhomogeneous random graphs in which vertices are assigned i.i.d.\ random weights, pairs of distinct vertices are connected by an edge independently with a probability that is a bi-variate function of the weights of the vertices, and single vertices are connected to themselves by a self-loop independently with a probability that is a uni-variate function of the weight of the vertex. We apply a renormalisation transformation in which vertices are aggregated into groups of equal size according to a greedy algorithm, namely, distinct groups of aggregated vertices are connected by an aggregated edge if and only if there is at least one edge connecting two constituent vertices across the groups, while a group of aggregated vertices is connected to itself by an aggregated self-loop if and only if there is at least one self-loop at an internal vertex or one edge connecting a pair of distinct internal vertices. We analyse what happens when the renormalisation transformation is iterated. In particular, we show that, starting from appropriately scaled connection functions, the iterated renormalised graphs converge to a two-parameter family of random graphs, acting as an attractor in a universality class. We consider a light-tailed regime, for which the scaling limit is a homogeneous Erdős--Rényi random graph, and a heavy-tailed regime, for which the scaling limit is an inhomogeneous random graph with stable infinite-mean random weights and an exponential disconnection function. Different scalings are needed for the two regimes. Which of the two regimes prevails depends on the choice of the connection functions and the choice of the law of the random weights.

math.PR↗

arXiv2026

Large deviation principle for friendship-biases in Galton--Watson trees

In this paper we consider the friendship-bias of the vertices in an infinite rooted Galton--Watson tree. The friendship-bias of a vertex is the difference between the average degree of the neighbours of the vertex and the degree of the vertex itself. A vertex is said to be of type $χ\in S$, with $S = \{-,0,+\}$, when its friendship-bias is, respectively, strictly negative, zero or strictly positive. We consider the fractions $f_l^χ$ of vertices of type $χ\in S$ along a random downward path up to branching depth $l \in \mathbb{N}$ and derive a large deviation principle (LDP) for the triple $(f_l^χ)_{χ\in S}$ as $l\to\infty$. The branching depth of a vertex counts the number of branchings that occur along the path that connects the vertex to the root of the tree. The rate in the LDP is $l$, while the rate function in the LDP is identified in terms of a variational formula minimising a relative entropy under a linear constraint. We focus on the case of binary branching, for which the rate function is already quite involved. We identify the qualitative properties of the rate function and show how it can be computed numerically. We briefly indicate how to proceed for more general branching and for vertex types along a tree consisting of a finite number of random downward paths. Our paper is the first to consider large deviations of vertex types.

math.PR↗

arXiv2026

Network evolution with self-reinforcement

We study a new class of preferential attachment trees with \emph{self-reinforcement}. At each time, each vertex is assigned a weight equal to the cumulative sum over past times of an affine function of its degree. A new vertex attaches itself via a single edge to an already present vertex with a probability proportional to the current weight of that vertex. This ``integrated popularity'' rule builds long memory directly into the attachment mechanism, thereby destroying the Markov and partial-exchangeability features that underlie the classical analysis of affine preferential attachment models. More broadly, the model connects to applied-probability work on long-memory self-interacting processes (such as the elephant random walk), emphasizing how non-Markovian reinforcement reshapes asymptotic behaviour. Despite this loss of structure, we identify an explicit exponent $ϕ=ϕ(δ)$ governing both local and global growth: typical degrees at time $n$ scale as $n^{1/ϕ}$, and the empirical degree distribution converges to a power-law with a tail exponent $ϕ+1$. We further prove Benjamini--Schramm local convergence to an infinite random rooted tree characterized via an embedded continuous-time branching process. The limiting tree is a \texttt{sin}-tree, and is \emph{not} the Pólya-type limiting tree arising in the non-reinforced setting. Our results provide a tractable probabilistic description of a natural ``memoryful'' network-growth mechanism, and quantify precisely how reinforcement renormalizes the classical preferential-attachment exponents.

math.PR↗

arXiv2026

Co-evolving vertex and edge dynamics in dense graphs

We consider a random graph in which vertices can have one of two possible colours. Each vertex switches its colour at a rate that is proportional to the number of vertices of the other colour to which it is connected by an edge. Each edge turns on or off according to a rate that depends on whether the vertices at its two endpoints have the same colour or not. We prove that, in the limit as the graph size tends to infinity and the graph becomes dense, the graph process converges, in a suitable path topology, to a limiting Markov process that lives on a certain subset of the space of coloured graphons. In the limit, the density of each vertex colour evolves according to a Fisher-Wright diffusion driven by the density of the edges, while the underlying edge connectivity structure evolves according to a stochastic flow whose drift depends on the densities of the two vertex colours.

math.PR↗

arXiv2026

The Widom-Rowlinson model: Mesoscopic fluctuations for the critical droplet

We study the critical droplet for a close-to-equilibrium Widom-Rowlinson model of interacting particles, represented by disks of radius $1$, in the two-dimensional plane at low temperature. The critical droplet is the set of macroscopic states that correspond to saddle points for the passage from a low-density supersaturated vapour to a stable high-density liquid. We analyse the mesoscopic fluctuations of the surface of the critical droplet, which turns out to be the set of particle configurations that are close to a disk of a certain deterministic radius. Our results represent the first detailed rigorous analysis of the surface fluctuations of a continuum interacting particle system exhibiting condensation and, as such, constitute a fundamental step in the study of phase separation from the perspective of stochastic geometry. At the same time, our results serve as a basis for the study of a non-equilibrium version of the Widom-Rowlinson model, to be analysed elsewhere, where they lead to a correction term in the Arrhenius formula for the average vapour-liquid crossover time.

math-ph↗

arXiv2026

Self-repellent branching random walk

We consider a system of particles performing a discrete-time binary branching random walk with independent standard normal increments subject to a penalty $\b$ for every pair of particles that get within distance $\e$ of each other at every time. We study the optimal configurations that minimise the sum of the spread out cost and the repulsion cost up to a given time horizon $N$. We show that at time $N$ particles are spread out over a distance $\asymp (\b\e)^{1/3} 2^{2N/3}$. We also show that the total cost of the optimal configurations up to time $N$ is $\asymp (\b\e)^{2/3} 2^{4N/3}$.

math.PR↗

arXiv2026

On the principal eigenvectors of random Markov matrices

We analyze the invariant distributions of continuous-time and discrete-time random walks on randomly weighted complete digraphs. These distributions correspond to the principal left eigenvectors of the associated random Markov generators and kernels, viewed as random matrices. While much is known about the spectra of these matrices, relatively little is known about the principal left eigenvectors, which are delicate random objects for which no explicit form is known. We consider a broad class of such matrices obtained by associating random weights to the vertices and edges of the complete digraph. Our main result concerns the total variation distance between the invariant distribution of the continuous-time random walk and the distribution that is inversely proportional to the vertex weights. It states that, if the edge weights are i.i.d. with a finite $p$-th moment for some $p>4$, then this distance a.s. converges to zero as the number of vertices grows large, even when the vertex weights are heavy-tailed. We further answer a question of Bordenave, Caputo, and Chafaï by showing that, despite the dependence of the entries in the corresponding Markov kernel, its invariant distribution is asymptotically uniform a.s., so long as the edge weights have a finite second moment.

math.PR↗

arXiv2026

Infection models on dense dynamic random graphs

We consider Susceptible-Infected-Recovered (SIR) models on dense dynamic random graphs, in which the joint dynamics of vertices and edges are co-evolutionary, i.e., they influence each other bidirectionally. In particular, edges appear and disappear over time depending on the states of the two connected vertices, on how long they have been infected, and on the total density of susceptible and infected vertices. Our main results establish functional laws of large numbers for the densities of susceptible, infected, and recovered vertices, jointly with the underlying evolving random graphs in the graphon space. Our results are supported by simulations, which characterize the limiting size of the epidemics, i.e., the limiting density of susceptible vertices, and how the peak of the epidemics depends on the rate of the evolution of the underlying graph. The proofs of our main results rely on the careful construction of a mimicking process, obtained by approximating the two-way feedback interaction between vertex and edge dynamics with a mean-field type interaction, acting only as one-way feedback, that remains sufficiently close to the original co-evolutionary process. To treat the more general setting in which edge dynamics are affected by the proportions of susceptible and infected individuals, we introduce a methodological extension of existing techniques. We thus show that our model exhibits multiple epidemic peaks -- a phenomenon observed in real-world epidemics -- which can emerge in models that incorporate mutual feedback between vertex and edge dynamics.

math.PR↗

arXiv2026

Sparse random graphs with many triangles

In this paper we consider the Erdős-Rényi random graph in the sparse regime in the limit as the number of vertices $n$ tends to infinity. We are interested in what this graph looks like when it contains many triangles, in two settings. First, we derive asymptotically sharp bounds on the probability that the graph contains a large number of triangles. We show that conditionally on this event, with high probability the graph contains an almost complete subgraph, i.e., the triangles form a near-clique, and has the same local limit as the original Erdős-Rényi random graph. Second, we derive asymptotically sharp bounds on the probability that the graph contains a large number of vertices that are part of a triangle. If order $n$ vertices are in triangles, then the local limit (provided it exists) is different from that of the Erdős-Rényi random graph. Our results shed light on the challenges that arise in the description of real-world networks, which often are sparse, yet highly clustered, and on exponential random graphs, which often are used to model such networks.

math.PR↗

arXiv2026

The multi-level friendship paradox for sparse random graphs

In Hazra, den Hollander and Parvaneh (2025) we analysed the friendship paradox for sparse random graphs. For four classes of random graphs we characterised the empirical distribution of the friendship biases between vertices and their neighbours at distance $1$, proving convergence as $n\to\infty$ to a limiting distribution, with $n$ the number of vertices, and identifying moments and tail exponents of the limiting distribution. In the present paper we look at the multi-level friendship bias between vertices and their neighbours at distance $k \in \mathbb{N}$ obtained via a $k$-step exploration according to a backtracking or a non-backtracking random walk. We identify the limit of empirical distribution of the multi-level friendship biases as $n\to\infty$ and/or $k\to\infty$. We show that for non-backtracking exploration the two limits commute for a large class of sparse random graphs, including those that locally converge to a rooted Galton-Watson tree. In particular, we show that the same limit arises when $k$ depends on $n$, i.e., $k=k_n$, provided $\lim_{n\to\infty} k_n = \infty$ under some mild conditions. We exhibit cases where the two limits do not commute and show the relevance of the mixing time of the exploration.

math.PR↗

arXiv2025

Homogeneous nucleation for two-dimensional Kawasaki dynamics

This is the third in a series of three papers in which we study a lattice gas subject to Kawasaki dynamics at inverse temperature $β>0$ in a large finite box $Λ_β\subset \mathbb{Z}^2$ whose volume depends on $β$. Each pair of neighbouring particles has a negative binding energy $-U<0$, while each particle has a positive activation energy $Δ>0$. The initial configuration is drawn from the grand-canonical ensemble restricted to the set of configurations where all the droplets are subcritical. Our goal is to describe, in the metastable regime $Δ\in (U,2U)$ and in the limit as $β\to\infty$, how and when the system nucleates, i.e., creates a critical droplet somewhere in $Λ_β$ that subsequently grows by absorbing particles from the surrounding gas. In the first paper we showed that subcritical droplets behave as quasi-random walks. In the second paper we used the results in the first paper to analyse how subcritical droplets form and dissolve on multiple space-time scales when the volume is moderately large, namely, $|Λ_β| = \mathrm{e}^{θβ}$ with $Δ< θ< 2Δ-U$. In the present paper we consider the setting where the volume is very large, namely, $|Λ_β| = \mathrm{e}^{Θβ}$ with $Θ< Γ-(2Δ-U)$, where $Γ$ is the energy of the critical droplet in the local model with fixed volume, and use the results in the first two papers to identify the nucleation time and the tube of typical trajectories towards nucleation. We will see that in a very large volume critical droplets appear more or less independently in boxes of moderate volume, a phenomenon referred to as homogeneous nucleation. One of the key ingredients in the proof is an estimate showing that no information can travel between these boxes on relevant time scales.

math.PR↗

arXiv2025

Optimal Structure of Signal Networks for Efficient Information Aggregation

This paper develops a mathematical framework to study signal networks, in which nodes can be active or inactive, and their activation or deactivation is driven by external signals and the states of the nodes to which they are connected via links. The focus is on determining the optimal number of key nodes (= highly connected and structurally important nodes) required to represent the global activation state of the network accurately. Motivated by neuroscience, medical science, and social science examples, we describe the node dynamics as a continuous-time inhomogeneous Markov process. Under mean-field and homogeneity assumptions, appropriate for large scale-free and disassortative signal networks, we derive differential equations characterising the global activation behaviour and compute the expected hitting time to network triggering. Analytical and numerical results show that two or three key nodes are typically sufficient to approximate the overall network state well, balancing sensitivity and robustness. Our findings provide insight into how natural systems can efficiently aggregate information by exploiting minimal structural components.

math.PR↗

arXiv2025

Tame sparse exponential random graphs

In this paper, we obtain a precise estimate of the probability that the sparse binomial random graph contains a large number of vertices in a triangle. The estimate of log of this probability is correct up to second order, and enables us to propose an exponential random graph model based on the number of vertices in a triangle. Specifically, by tuning a single parameter, we can with high probability induce any given fraction of vertices in a triangle. Moreover, in the proposed exponential random graph model we derive the large deviation principle for the number of edges. As a byproduct, we propose a consistent estimator of the tuning parameter.

math.PR↗

arXiv2025

Self-Reinforced Preferential Attachment

We consider a preferential attachment random graph with self-reinforcement. Each time a new vertex comes in, it attaches itself to an old vertex with a probability that is proportional to the sum of the degrees of that old vertex at all prior times. The resulting growing graph is a random tree whose vertices have degrees that grow polynomially fast in time. We compute the growth exponent, show that it is strictly larger than the growth exponent in the absence of self-reinforcement, and develop insight into how the self-reinforcement affects the growth. Proofs are based on a stochastic approximation scheme.

math.PR↗

arXiv2025

The Triangle Friendship Paradox

We consider the generalised friendship paradox, focussing on the number of triangles at a vertex as the relevant attribute. We show that, contrary to the setting where the attribute is the number of edges at a vertex or the number of wedges at a vertex, the average friendship-bias of the number of triangles at a vertex is not always non-negative. We identify classes of finite deterministic graphs for which the bias is non-negative, and provide examples of finite deterministic graphs for which it is not. For certain classes of sparse and dense random graphs, we compute the scaling of the bias in the limit as the number of vertices tends to infinity.

math.PR↗

arXiv2025

Continuum graph dynamics via population dynamics: well-posedness, duality and equilibria

This paper introduces graphemes for constructing and analyzing stochastic processes that describe the evolution of large dynamic graphs. Unlike graphons, which capture the static properties of dense graphs via exchangeability or subgraph densities, graphemes are capable of modeling the full space-time evolution of graphs. A grapheme is an equivalence class of triples: (Polish space, symmetric {0,1}-valued connection function, sampling probability measure). We focus on embeddings in ultrametric spaces, encoding the graph history and linking directly to population dynamics models. Graphemes utilize stronger equivalences (homeomorphism, isometry) than graphons. We construct grapheme-valued Markov processes as limits of finite graph evolutions, driven by Fleming-Viot, Dawson-Watanabe, and McKean-Vlasov analogues. We establish characterization via well-posed martingale problems, yielding strong Markov processes with the Feller property and continuous paths (diffusions). Duality relations involving coalescent processes are derived. We identify non-trivial equilibria, linked to classical distributions from population genetics. This framework extends [arXiv:1908.06241] by incorporating history, enabling rigorous analysis via martingale problems, and characterizing non-trivial long-term behavior.

math.PR↗

arXiv2025

The friendship paradox for sparse random graphs

Let $G_n$ be an undirected finite graph on $n\in\mathbb{N}$ vertices labelled by $[n] = \{1,\ldots,n\}$. For $i \in [n]$, let $Δ_{i,n}$ be the friendship bias of vertex $i$, defined as the difference between the average degree of the neighbours of vertex $i$ and the degree of vertex $i$ itself when $i$ is not isolated, and zero when $i$ is isolated. Let $μ_n$ denote the friendship-bias empirical distribution, i.e., the measure that puts mass $\frac{1}{n}$ at each $Δ_{i,n}$, $i \in [n]$. The friendship paradox says that $\int_{\mathbb{R}} xμ_n(\mathrm{d}x) \geq 0$, with equality if and only if in each connected component of $G_n$ all the degrees are the same. We show that if $(G_n)_{n\in\mathbb{N}}$ is a sequence of sparse random graphs that converges to a rooted random tree in the sense of convergence locally in probability, then $μ_n$ converges weakly to a limiting measure $μ$ that is expressible in terms of the law of the rooted random tree. We study $μ$ for four classes of sparse random graphs: the homogeneous Erdős-Rényi random graph, the inhomogeneous Erdős-Rényi random graph, the configuration model and the preferential attachment model. In particular, we compute the first two moments of $μ$, identify the right tail of $μ$, and argue that $μ([0,\infty))\geq\tfrac{1}{2}$, a property we refer to as friendship paradox significance.

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