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

TL;DR

This paper investigates the asymptotic location of high-degree vertices in weighted recursive graphs with bounded random weights, confirming a conjecture and extending known results from simpler models to more general graph structures.

Contribution

It establishes the existence of a critical exponent for the label size of maximum degree vertices in WRGs and extends prior results from RRT to weighted recursive graphs.

Findings

Existence of a critical exponent _m for label size growth.

Almost sure convergence of label size to n^{_m} for maximum degree vertices.

Joint convergence of degree and label for high-degree vertices in WRT models.

Abstract

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.