arXiv subjects

Serte Donderwinkel

Publications and source records attributed to Serte Donderwinkel.

23 records · Page 2Linked to original sources

arXiv2024

Random friend trees

We study a random recursive tree model featuring complete redirection called the random friend tree and introduced by Saramäki and Kaski. Vertices are attached in a sequential manner one by one by selecting an existing target vertex and connecting to one of its neighbours (or friends), chosen uniformly at random. This model has interesting emergent properties, such as a highly skewed degree sequence. In contrast to the preferential attachment model, these emergent phenomena stem from a local rather than a global attachment mechanism. The structure of the resulting tree is also strikingly different from both the preferential attachment tree and the uniform random recursive tree: every edge is incident to a macro-hub of asymptotically linear degree, and with high probability all but at most $n^{9/10}$ vertices in a tree of size $n$ are leaves. We prove various results on the neighbourhood of fixed vertices and edges, and we study macroscopic properties such as the diameter and the degree distribution, providing insights into the overall structure of the tree. We also present a number of open questions on this model and related models.

math.PR↗

arXiv2024

Random trees have height $O(\sqrt{n})$

We obtain new non-asymptotic tail bounds for the height of uniformly random trees with a given degree sequence, simply generated trees and conditioned Bienaymé trees (the family trees of branching processes), in the process settling three conjectures of Janson (2012) and answering several other questions from the literature. Moreover, we define a partial ordering on degree sequences and show that it induces a stochastic ordering on the heights of uniformly random trees with given degree sequences. The latter result can also be used to show that sub-binary random trees are stochastically the tallest trees with a given number of vertices and leaves (and thus that random binary trees are the stochastically tallest random homeomorphically irreducible trees with a given number of vertices). Our proofs are based in part on the bijection between trees and sequences introduced by Foata and Fuchs (1970), which can be recast to provide a line-breaking construction of random trees with given vertex degrees as shown in Addario-Berry, Blanc-Renaudie, Donderwinkel, Maazoun and Martin (2023).

math.PR↗

arXiv2022

The Foata-Fuchs proof of Cayley's formula, and its probabilistic uses

We present a very simple bijective proof of Cayley's formula due to Foata and Fuchs (1970). This bijection turns out to be very useful when seen through a probabilistic lens; we explain some of the ways in which it can be used to derive probabilistic identities, bounds, and growth procedures for random trees with given degrees, including random d-ary trees. We also introduce a partial order on the degree sequences of rooted trees, and conjecture that it induces a stochastic partial order on heights of random rooted trees with given degrees.

math.CO↗

arXiv2021

Convergence of the height process of supercritical Galton-Watson forests with an application to the configuration model in the critical window

We show joint convergence of the Lukasiewicz path and height process for slightly supercritical Galton-Watson forests. This shows that the height processes for supercritical continuous state branching processes as constructed by Lambert (2002) are the limit under rescaling of their discrete counterparts. Unlike for (sub-)critical Galton-Watson forests, the height process does not encode the entire metric structure of a supercritical Galton-Watson forest. We demonstrate that this result is nonetheless useful, by applying it to the configuration model with an i.i.d. power-law degree sequence in the critical window, of which we obtain the metric space scaling limit in the product Gromov-Hausdorff-Prokhorov topology, which is of independent interest.

math.PR↗

arXiv2021

Universality for the directed configuration model: metric space convergence of the strongly connected components at criticality

We consider the strongly connected components (SCCs) of a uniform directed graph on $n$ vertices with i.i.d. in- and out-degree pairs distributed as $(D^-,D^+)$, with $\mathbb E[D^+]=\mathbb E[D^-]=μ$. We condition on equal total in- and out-degree. A phase transition for the emergence of a giant SCC is known to occur at the critical value $\mathbb E[D^-D^+] = μ$. We study the model at this critical value and, additionally, require that $\mathbb E[(D^-)^i(D^+)^j]<\infty$ for all $i+j\leq 3$, and for $(i,j)=(1,3)$ and $(i,j)=(3,1)$. We show that, under these conditions, the SCCs ranked by decreasing number of edges with distances rescaled by $n^{-1/3}$ converge in distribution to a sequence of finite strongly connected directed multigraphs with edge lengths, and that these are either $3$-regular or loops. The limit objects lie in a $3$-parameter family, which contains the scaling limit of the SCCs in the directed Erdős-Rényi model at criticality as found by Goldschmidt and Stephenson (2019). This is the first universality result for the scaling limit of a critical directed graph model and the first quantitative result on the directed configuration model at criticality. As a trivial consequence, the largest SCCs at criticality contain $Θ(n^{1/3})$ vertices and edges in probability, and the diameter of the directed graph at criticality is $Ω(n^{1/3})$ in probability. We use a metric on the space of weighted multigraphs in which two multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge lengths. We use the product topology on the sequence of multigraphs. Our method of proof involves a depth-first exploration of the directed graph, resulting in a spanning forest with additional identifications, of which we study the limit under rescaling.

math.PR↗