The location of high-degree vertices in weighted recursive graphs with bounded random weights
Bas Lodewijks
TL;DR
For the Weighted Recursive Tree (WRT) model, that is, the WRG model with $m=1, it is proved the joint convergence of the rescaled degree and label of high-degree vertices under additional assumptions on the vertex-weight distribution.
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 $μ_m$, such that the typical label size of the maximum degree vertex equals $n^{μ_m(1+o(1))}$ 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.