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Bas Lodewijks

Publications and source records attributed to Bas Lodewijks.

At least 19 recordsLinked to original sources
arXiv2026

High-degree vertices in uniform recursive directed acyclic graphs with freezing

We study uniform recursive directed acyclic graphs with freezing. Here, a graph is built by adding vertices one-by-one and connecting a new vertex to $m\in\mathbb N$ uniformly selected vertices already present. At certain steps vertices can also be frozen, and arriving vertices are not allowed to connect to frozen vertices. This model generalises the uniform attachment tree with freezing, introduced by Bellin et. al (which corresponds to the case $m=1$) as well as the uniform recursive directed acyclic graph model (where no vertices are frozen). Under mild assumptions on when vertices are frozen, we study the empirical degree distribution, large degrees in the graph, and other properties of large-degree vertices such as their label and distance to the first vertex in the graph. Our work improves and/or extends various results from the literature on uniform attachment trees (with freezing) and uniform recursive directed acyclic graphs without freezing. In particular, our results show that statistics that are determined `locally' (e.g. the empirical degree distribution and maximum degree) are essentially unaffected by the freezing of vertices, whereas statistics that are determined `globally' (e.g. the length of paths between vertices) are highly affected by introducing freezing. The analysis relies on adapting the Kingman coalescent construction for uniform attachment trees to the non-tree setting.

math.PR↗
arXiv2026

On the depth of depth-weighted trees

The depth-weighted tree DWT($f$) with weight function $f:\{0,1,2,\ldots\}\to (0,\infty)$ is a dynamic random tree grown from a root $r$ where vertices arrive consecutively and every new vertex attaches to a parent $u$ with probability proportional to $f$(distance between $u$ and $r$). This work is dedicated to a systematic analysis of the depth of DWT($f$). Namely, we provide precise analytic expressions of the typical depth of DWT($f$) for convergent, periodic, slowly growing, and (super-)exponentially growing weight functions. Furthermore, for bounded or exponentially growing $f$, we determine the typical depth up to a multiplicative constant, thus confirming and strengthening a conjecture of Leckey, Mitsche and Wormald.

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arXiv2025

Local criteria for global connectivity comparisons: beyond stochastic domination

We introduce a site-wise domination criterion for local percolation models, which enables the comparison of one-arm probabilities even in the absence of stochastic domination. The method relies on a local-to-global principle: if, at each site, one model is more likely than the other to connect to a subset of its neighbors, for all nontrivial such subsets, then this advantage propagates to connectivity events at all scales. In this way, we obtain a robust alternative to stochastic domination, applicable in all cases where the latter works and in many where it does not. As a main application, we compare classical Bernoulli bond percolation with degree-constrained models, showing that degree constraints enhance percolation, and obtain asymptotically optimal bounds on critical parameters for degree-constrained models.

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arXiv2025

Preferential Attachment Trees with Vertex Death: Persistence of the Maximum Degree

We consider an evolving random discrete tree model called Preferential Attachment with Vertex Death, as introduced by Deijfen. Initialised with an alive root labelled $1$, at each step $n\geq1$ either a new vertex with label $n+1$ is introduced that attaches to an existing alive vertex selected preferentially according to a function $b$, or an alive vertex is selected preferentially according to a function $d$ and killed. In this article we introduce a generalised concept of persistence for evolving random graph models. Let $O_n$ be the smallest label among all alive vertices (the oldest alive vertex), and let $I_n^m$ be the label of the alive vertex with the $m^{\mathrm{th}}$ largest degree. We say a persistent $m$-hub exists if $I_n^m$ converges almost surely, we say that persistence occurs when $I_n^1/O_n$ is tight, and that lack of persistence occurs when $I_n^1/O_n$ tends to infinity. We identify two regimes called the infinite lifetime and finite lifetime regimes. In the infinite lifetime regime, vertices are never killed with positive probability. Here, we provide conditions under which we prove the (non-)existence of persistent $m$-hubs for any $m\in\mathbb N$. This expands and generalises recent work of Iyer, which covers the case $d\equiv 0$ and $m=1$. In the finite lifetime regime, vertices are killed after a finite number of steps almost surely. Here we provide conditions under which we prove the occurrence of persistence, which complements recent work of Heydenreich and the author, where lack of persistence is studied for preferential attachment with vertex death.

math.PR↗
arXiv2025

Preferential Attachment Trees with Vertex Death: Lack of Persistence of the Maximum Degree

We consider an evolving random discrete tree model called Preferential Attachment with Vertex Death, as introduced by Deijfen. Initialised with an alive root labelled $1$, at each step $n\geq1$ either a new vertex with label $n+1$ is introduced that attaches to an existing alive vertex selected preferentially according to a function $b$, or an alive vertex is selected preferentially according to a function $d$ and killed. We introduce a generalised concept of persistence for evolving random graph models. Let $O_n$ be the smallest label among all alive vertices (the oldest alive vertex), and let $I_n$ be the label of the alive vertex with the largest degree (among all alive vertices). Persistence occurs when $I_n/O_n$ is tight; lack of persistence occurs when $I_n/O_n$ diverges with $n$. We study lack of persistence and identify two regimes: the old are rich and the rich die young regime. In the rich are old regime, though the oldest alive vertices in the tree typically have the largest degrees, lack of persistence can occur subject to the condition $\sum_{i=0}^\infty 1/(b(i)+d(i))^2=\infty$, under which lucky vertices that are younger than the oldest vertices can attain the largest degrees by step $n$, generalising results by Banerjee and Bhamidi. In contrast, lack of persistence always occurs in the rich die young regime. This regime is novel and cannot be observed in models without death. Here, vertices can survive exceptionally long by obtaining a low degree, whereas vertices with a large degree die much faster, causing lack of persistence. A main technique is an embedding of the discrete tree process into a Crump-Mode-Jagers branching process and a higher-order analysis of the resulting birth-death mechanism based on moderate deviation principles with exponential tilting.

math.PR↗
arXiv2024

A star is born: Explosive Crump-Mode-Jagers branching processes

We study a family of Crump--Mode--Jagers branching processes in random environment that explode, i.e. that grow infinitely large in finite time with positive probability. Building on recent work of the author and Iyer (``On the structure of genealogical trees associated with explosive Crump--Mode--Jagers branching processes", arXiv:2311.14664, 2023), we weaken certain assumptions required to prove that the branching process, at the time of explosion, contains a (unique) individual with infinite offspring. We then apply these results to super-linear preferential attachment models. In particular, we fill gaps in some of the cases analysed in Appendix A of the work of the author and Iyer and study a large range of previously unattainable cases.

math.PR↗
arXiv2024

Long-range competition on the torus

We study competition between two growth models with long-range correlations on the torus $\mathbb T_n^d$ of size $n$ in dimension $d$. We append the edge set of the torus $\mathbb T_n^d$ by including all non-nearest-neighbour edges, and from two source vertices $v^\ominus$ and $v^\oplus$ in $\mathbb T_n^d$ two infection processes $\ominus$ and $\oplus$ start spreading to other vertices. Each susceptible vertex can be infected by at most one infection type and when infected stays infected forever (i.e.\ competing SI models). A vertex $v$ infected with type $\square\in\{\ominus,\oplus\}$ infects a susceptible vertex $u$ at rate $\lambda_\square \|u-v\|^{-\alpha_\square}$, where $\lambda_{\ominus}=\lambda_\ominus(n),\lambda_\oplus=\lambda_\oplus(n)>0$ and $\alpha_\ominus=\alpha_\ominus(n),\alpha_\oplus=\alpha_\oplus(n)\in[0,d)$ are allowed to depend on $n$. We study \emph{coexistence}, the event that both infections reach an asymptotically positive proportion of the graph as $n$ tends to infinity, and identify precisely when coexistence occurs. In the case of absence of coexistence, we outline several phase transitions in the size of the infection that reaches a negligible proportion of the vertices, which depends on the ratio of the sum of infection rates across all vertices of type $\ominus$ and $\oplus$. The work extends known results for the case $\alpha_\ominus(n)=\alpha_\oplus(n)\equiv 0$ and $\lambda_\ominus(n)\equiv 1, \lambda_\oplus(n)\equiv \lambda>0$, and includes general and novel results that cannot be observed when the model parameters are fixed and independent of $n$. The main technical contribution is a coupling of the competition process with branching random walks, where we are able to use the coupling even when the coupling error between the competition process and the branching random walks is of the same order of magnitude as the size of the coupled processes.

math.PR↗
arXiv2023

Long-range first-passage percolation on the torus

We study a geometric version of first-passage percolation on the complete graph, known as long-range first-passage percolation. Here, the vertices of the complete graph $\mathcal K_n$ are embedded in the $d$-dimensional torus $\mathbb T_n^d$, and each edge $e$ is assigned an independent transmission time $T_e=\|e\|_{\mathbb T_n^d}^\alpha E_e$, where $E_e$ is a rate-one exponential random variable associated with the edge $e$, $\|\cdot\|_{\mathbb T_n^d}$ denotes the torus-norm, and $\alpha\geq0$ is a parameter. We are interested in the case $\alpha\in[0,d)$, which corresponds to the instantaneous percolation regime for long-range first-passage percolation on $\mathbb Z^d$ studied by Chatterjee and Dey, and which extends first-passage percolation on the complete graph (the $\alpha=0$ case) studied by Janson. We consider the typical distance, flooding time, and diameter of the model. Our results show a $1,2,3$-type result, akin to first-passage percolation on the complete graph as shown by Janson. The results also provide a quantitative perspective to the qualitative results observed by Chatterjee and Dey on $\mathbb Z^d$.

math.PR↗
arXiv2023

On the structure of genealogical trees associated with explosive Crump-Mode-Jagers branching processes

We study the structure of genealogical trees associated with explosive Crump-Mode-Jagers branching processes (stopped at the explosion time), proving criteria for the associated tree to contain a node of infinite degree (a star) or an infinite path. Next, we provide uniqueness criteria under which with probability $1$ there exists exactly one of a unique star or a unique infinite path. Under the latter uniqueness criteria, we also provide an example where, with strictly positive probability less than $1$, there exists a unique node of infinite degree in the model, thus this probability is not restricted to being $0$ or $1$. Moreover, we provide structure theorems when there is a star, when certain trees appear as sub-trees of the star infinitely often. We apply our results to general discrete evolving tree models of explosive recursive trees with fitness, and as particular cases, we study a family of super-linear preferential attachment models with fitness. In the latter regime, we derive phase transitions in the model parameters in three different examples, leading to either exactly one star with probability $1$, or one infinite path with probability $1$, with every node having finite degree. Furthermore, we highlight examples where sub-trees $T$ of arbitrary size can appear infinitely often; behaviour that is markedly distinct from super-linear preferential attachment models studied in the literature so far.

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arXiv2023

Percolation in lattice $k$-neighbor graphs

We define a random graph obtained via connecting each point of $\mathbb{Z}^d$ independently to a fixed number $1 \leq k \leq 2d$ of its nearest neighbors via a directed edge. We call this graph the directed $k$-neighbor graph. Two natural associated undirected graphs are the undirected and the bidirectional $k$-neighbor graph, where we connect two vertices by an undirected edge whenever there is a directed edge in the directed $k$-neighbor graph between them in at least one, respectively precisely two, directions. In these graphs we study the question of percolation, i.e., the existence of an infinite self-avoiding path. Using different kinds of proof techniques for different classes of cases, we show that for $k=1$ even the undirected $k$-neighbor graph never percolates, but the directed one percolates whenever $k \geq d+1$, $k \geq 3$ and $d \geq 5$, or $k \geq 4$ and $d=4$. We also show that the undirected $2$-neighbor graph percolates for $d=2$, the undirected $3$-neighbor graph percolates for $d=3$, and we provide some positive and negative percolation results regarding the bidirectional graph as well. A heuristic argument for high dimensions indicates that this class of models is a natural discrete analogue of the $k$-nearest-neighbor graphs studied in continuum percolation, and our results support this interpretation.

math.PR↗
arXiv2023

On joint properties of vertices with a given degree or label in the random recursive tree

In this paper, we study the joint behaviour of the degree, depth and label of and graph distance between high-degree vertices in the random recursive tree. We generalise the results obtained by Eslava and extend these to include the labels of and graph distance between high-degree vertices. The analysis of both these two properties of high-degree vertices is novel, in particular in relation to the behaviour of the depth of such vertices. In passing, we also obtain results for the joint behaviour of the degree and depth of and graph distance between any fixed number of vertices with a prescribed label. This combines several isolated results on the degree and depth of and graph distance between vertices with a prescribed label already present in the literature. Furthermore, we extend these results to hold jointly for any number of fixed vertices and improve these results by providing more detailed descriptions of the distributional limits. Our analysis is based on a correspondence between the random recursive tree and a representation of the Kingman $n$-coalescent.

math.PR↗
arXiv2023

Fine asymptotics for the maximum degree in weighted recursive trees with bounded random weights

A weighted recursive tree is an evolving tree in which vertices are assigned random vertex-weights and new vertices connect to a predecessor with a probability proportional to its weight. Here, we study the maximum degree and near-maximum degrees in weighted recursive trees when the vertex-weights are almost surely bounded and their distribution function satisfies a mild regularity condition near zero. We are able to specify higher-order corrections to the first order growth of the maximum degree established in prior work. The accuracy of the results depends on the behaviour of the weight distribution near the largest possible value and in certain cases we manage to find the corrections up to random order. Additionally, we describe the tail distribution of the maximum degree, the distribution of the number of vertices attaining the maximum degree, and establish asymptotic normality of the number of vertices with near-maximum degree. Our analysis extends the results proved for random recursive trees (where the weights are constant) to the case of random weights. The main technical result shows that the degrees of several uniformly chosen vertices are asymptotically independent with explicit error corrections.

math.PR↗
arXiv2022

Dynamic random graphs with vertex removal

We introduce and analyse a Dynamic Random Graph with Vertex Removal (DRGVR) defined as follows. At every step, with probability $p > 1/2$ a new vertex is introduced, and with probability $1-p$ a vertex, chosen uniformly at random among the present ones (if any), is removed from the graph together with all edges adjacent to it. In the former case, the new vertex connects by an edge to every other vertex with probability inversely proportional to the number of vertices already present. We prove that the DRGVR converges to a local limit and determine this limit. Moreover, we analyse its component structure and distinguish a subcritical and a supercritical regime with respect to the existence of a giant component. As a byproduct of this analysis, we obtain upper and lower bounds for the critical parameter. Furthermore, we provide precise expression of the maximum degree (as well as in- and out-degree for a natural orientation of the DRGVR). Several concentration and stability results complete the study.

math.PR↗
arXiv2022

On the first and second largest components in the percolated Random Geometric Graph

The percolated random geometric graph $G_n(\lambda, p)$ has vertex set given by a Poisson Point Process in the square $[0,\sqrt{n}]^2$, and every pair of vertices at distance at most 1 independently forms an edge with probability $p$. For a fixed $p$, Penrose proved that there is a critical intensity $\lambda_c = \lambda_c(p)$ for the existence of a giant component in $G_n(\lambda, p)$. Our main result shows that for $\lambda > \lambda_c$, the size of the second-largest component is a.a.s. of order $(\log n)^2$. Moreover, we prove that the size of the largest component rescaled by $n$ converges almost surely to a constant, thereby strengthening results of Penrose. We complement our study by showing a certain duality result between percolation thresholds associated to the Poisson intensity and the bond percolation of $G(\lambda, p)$ (which is the infinite volume version of $G_n(\lambda,p)$). Moreover, we prove that for a large class of graphs converging in a suitable sense to $G(\lambda, 1)$, the corresponding critical percolation thresholds converge as well to the ones of $G(\lambda,1)$.

math.PR↗
arXiv2021

The location of high-degree vertices in weighted recursive graphs with bounded random weights

We study the asymptotic growth rate of the label size of high-degree vertices in weighted recursive graphs (WRG) when the weights are i.i.d. almost surely bounded random variables, and as a result confirm a conjecture by Lodewijks and Ortgiese. WRGs are a generalisation of the random recursive tree (RRT) and directed acyclic graph model (DAG), in which vertices are assigned vertex-weights and where new vertices attach to $m\in\mathbb N$ predecessors, each selected independently with a probability proportional to the vertex-weight of the predecessor. Prior work established the asymptotic growth rate of the maximum degree of the WRG model and here we show that there exists a critical exponent $\mu_m$, such that the typical label size of the maximum degree vertex equals $n^{\mu_m(1+o(1))}$ almost surely as $n$, the size of the graph, tends to infinity. These results extend and improve on the asymptotic behaviour of the location of the maximum degree, formerly only known for the RRT model, to the more general weighted multigraph case of the WRG model. Moreover, for the Weighted Recursive Tree (WRT) model, that is, the WRG model with $m=1$, we prove the joint convergence of the rescaled degree and label of high-degree vertices under additional assumptions on the vertex-weight distribution, and also extend results on the growth rate of the maximum degree obtained by Eslava, Lodewijks, and Ortgiese.

math.PR↗
arXiv2020

The maximal degree in random recursive graphs with random weights

We study a generalisation of the random recursive tree (RRT) model and its multigraph counterpart, the uniform directed acyclic graph (DAG). Here, vertices are equipped with a random vertex-weight representing initial inhomogeneities in the network, so that a new vertex connects to one of the old vertices with a probability that is proportional to their vertex-weight. We first identify the asymptotic degree distribution of a uniformly chosen vertex for a general vertex-weight distribution. For the maximal degree, we distinguish several classes that lead to different behaviour: For bounded vertex-weights we obtain results for the maximal degree that are similar to those observed for RRTs and DAGs. If the vertex-weights have unbounded support, then the maximal degree has to satisfy the right balance between having a high vertex-weight and being born early. For vertex-weights in the Frechet maximum domain of attraction the first order behaviour of the maximal degree is random, while for those in the Gumbel maximum domain of attraction the leading order is deterministic. Surprisingly, in the latter case, the second order is random when considering vertices in a compact window in the optimal region, while it becomes deterministic when considering all vertices.

math.PR↗
arXiv2020

A phase transition for preferential attachment models with additive fitness

Preferential attachment models form a popular class of growing networks, where incoming vertices are preferably connected to vertices with high degree. We consider a variant of this process, where vertices are equipped with a random initial fitness representing initial inhomogeneities among vertices and the fitness influences the attractiveness of a vertex in an additive way. We consider a heavy-tailed fitness distribution and show that the model exhibits a phase transition depending on the tail exponent of the fitness distribution. In the weak disorder regime, one of the old vertices has maximal degree irrespective of fitness, while for strong disorder the vertex with maximal degree has to satisfy the right balance between fitness and age. Our methods use martingale methods to show concentration of degree evolutions as well as extreme value theory to control the fitness landscape.

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