Geometry of weighted recursive and affine preferential attachment trees

Abstract

We study two models of growing recursive trees. For both models, the tree initially contains a single vertex u1 and at each time n ≥ 2 a new vertex un is added to the tree and its parent is chosen randomly according to some rule. In the weighted recursive tree, we choose the parent u k of un among {u1, u2, . . . , un−1} with probability proportional to w k , where (wn) n≥1 is some deterministic sequence that we fix beforehand.In the affine preferential attachment tree with fitnesses, the probability of choosing any u k is proportional to a k + deg + (u k ), where deg + (u k ) denotes its current number of children, and the sequence of fitnesses (an) n≥1 is deterministic and chosen as a parameter of the model. We show that for any sequence (an) n≥1 , the corresponding preferential attachment tree has the same distribution as some weighted recursive tree with a random sequence of weights (with some explicit distribution). We then prove almost sure scaling limit convergences for some statistics associated with weighted recursive trees as time goes to infinity, such as degree sequence, height, profile and also the weak convergence of some measures carried on the tree. Thanks to the connection between the two models, these results also apply to affine preferential attachment trees.

Degree distributions in recursive trees with fitnesses

“…Interestingly, as observed in [5] and confirmed rigorously in [8,12,14], there is a critical condition on the weight distribution under which this model undergoes a phase transition, resulting in a Bose-Einstein condensation: in the limit, a positive fraction of vertices accumulate around vertices of maximum weight. In a similar model known as preferential attachment with additive fitness, introduced in [16] and studied mathematically in [3,23,34], vertices connect to previous vertices with probability proportional to the sum of their degree (or degree minus one) and their weight. A number of other interesting preferential attachment models with fitness have been studied, including a model of preferential attachment with both additive and multiplicative weights [16], a related continuous-time model which incorporates ageing of vertices [18], and discrete-time models with co-existing additive and multiplicative attachment rules [1,22].…”

“…It was also introduced independently by Janson in the case that all weights are one except at the root, motivated by applications to infinite-colour Pólya urns [21]. In [34], Sénizergues showed that a preferential attachment tree with additive fitness with deterministic weights is equal in distribution to an associated weighted random recursive tree with random weights, an interesting link between the two classes of models.…”

Periodic Pólya urns, the density method and asymptotics of Young tableaux

“…• in Peköz, Röllin, and Ross [71], as distributions of processes on walks, trees, urns, and preferential attachments in graphs, where these authors also consider what they call a Pólya urn with immigration, which is a special case of a periodic Pólya urn (other models or random graphs have these distributions as limit laws [20,85]);…”

Large deviations principle for a stochastic process with random reinforced relocations

“…random variables, under an 8-th moment condition. Sénizergues [15] proved results about its degree distribution and height for a deterministic sequence of weights under very general assumptions (with more refined results about the height -maximum distance of a node to the root-in Pain and Sénizergues [13]). Finally, Iyer [9] and Lodewijks and Ortgiese [11] prove asymptotic results on, respectively the degree distribution and the largest degree of the wrrt with i.i.d.…”

“…This result is independent from Theorem 1.5; we state it because it has the potential for independent interest since wrrts are studied in unrelated contexts (see, e.g. [3,15], which we discuss in more detail later).…”