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

Abstract

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. 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 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.

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

“…These results are novel even for the random recursive tree model for which a weaker convergence result (compared to Theorem 2.3) of the first-order asymptotic logarithmic behaviour of the maximum degree vertex location was already proved by Banerjee and Bhamidi in [1]. The main reason we are able to provide a more detailed result for the WRT model rather than the more general WRG model is the fact that the author, together with Eslava and Ortgiese, provided the necessary higher-order behaviour of the maximum degree for the WRT model in [5].…”

“…Another important element that is required to prove Theorem 2.6, as already mentioned in Section 2, is the higher-order behaviour of the maximum degree in the WRT model, as proved by Eslava, the author and Ortgiese in [5]. We state their results here for completeness.…”

“…In recent work by Eslava, Lodewijks and Ortgiese [5], more refined asymptotic behaviour of the maximum degree is presented for the weighted recursive tree model (WRT), that is, the WRG model with m = 1, under additional assumptions on the vertex-weight distribution. These more precise results would enable, to some extent, a simplification of the proof of Theorem 2.3 for the case m = 1 (though it would not be applicable in the general setting we present here).…”

“…Furthermore, under additional assumptions on the vertex-weight distribution, we are able to provide bounds on the fluctuations around the exponent γ 1 in the case m = 1, that is, for the WRT model. These assumptions are identical to the assumptions made by Eslava, Lodewijks and Ortgiese in [5] to provide higher-order asymptotic results for the growth rate of the maximum degree in the WRT model. The behaviour of the higher-order fluctuations is completely novel and was not considered by Banerjee and Bhamidi for the RRT model.…”

“…Finally, Eslava, Lodewijks and Ortgiese [5] describe the asymptotic behaviour of the maximum degree in the WRT model in more detail (compared to [10]) when the vertex-weights are i.i.d. bounded random variables, under additional assumptions on the vertex-weight distribution.…”