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

Abstract

We study the asymptotic logarithmic growth rate of the label size of vertices that attain the maximum degree in weighted recursive graphs (WRG) when the weights are independent, identically distributed, almost surely bounded random variables, and as a result confirm a conjecture by Lodewijks and Ortgiese [10]. WRGs are a generalisation of the random recursive tree (RRT) and directed acyclic graph model (DAG), in which vertices are assigned vertexweights and where new vertices attach to m ∈ N predecessors 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 study the asymptotic logarithmic growth rate of the location, that is, the label size of the vertices that attain the maximum degree. We show that there exists a critical exponent γm, such that the typical age/label size of the maximum degree vertex equals n γm(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, in the Weighted Recursive Tree (WRT) model, that is, the WRG model with m = 1, we are able to provide bounds on the fluctuations around the exponent γ 1 under additional assumptions on the vertex-weight distribution. These results are novel even for the RRT model. The approach in this paper combines a refined version of the approach developed for studying the maximum degree of the WRG model with more precise union bounds.

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

“…First, we study the joint behaviour of the depth and label of and graph distance between any finite number of vertices selected uniformly at random, conditionally on having a degree that exceeds a certain quantity. We combine, extend, improve and recover the results of the author [23] (in the particular case of the random recursive tree) and Eslava [12]. We also recover the results of Addario-Berry and Eslava [1] and Eslava, the author, and Ortgiese [14] (again, in the particular case of the random recursive tree).…”

“…of the points x ∈ P). We remark that the weak convergence of P (nj ) to P ε in M # Z * along subsequences has been established by Addario-Berry and Eslava in [1] (later generalised to weighted recursive trees by Eslava, the author and Ortgiese in [14] and extended to marked point processes by the author in [23]) and that Eslava established the weak convergence of MP (nj ) along subsequences, which is MP (nj ) with each mark restricted to the first element (i.e. not considering the label), in [12].…”

“…Proof of Proposition 3.1. The proof essentially follows the same approach as the proof of [23,Proposition 5.4]. However, as certain estimations and definitions differ, we include it here for completeness.…”

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

“…First, we study the joint behaviour of the depth and label of and graph distance between any fixed number of vertices selected uniformly at random, conditionally on having a degree that exceeds a certain quantity. We combine, extend, improve and recover the results of the author [23] (in the particular case of the random recursive tree) and Eslava [12]. We also recover the results of Addario-Berry and Eslava [1] and Eslava, the author, and Ortgiese [14] (again, in the particular case of the random recursive tree).…”

“…As far as the author is aware, only a handful of papers consider the joint behaviour of different statistics for the random recursive tree. In [12], Eslava studies the depth of high-degree vertices, Banerjee and Bhamidi study the label size of the vertex attaining the maximum degree in [3], and the author studies the labels of high-degree vertices in the more general weighted recursive tree model [23], of which the random recursive tree model is a particular example.…”