Leaves of preferential attachment trees

We provide a local probabilistic description of the limiting statistics of large preferential attachment trees in terms of the ordinary degree (number of neighbors) but augmented with information on leafdegree (number of neighbors that are leaves). The full description is the joint degree-leafdegree distribution $n_{k,\ell}$, which we derive from its associated multivariate generating function. From $n_{k,\ell}$ we obtain the leafdegree distribution, $m_{\ell}$, as well as the fraction of vertices that are protected (nonleaves with leafdegree zero) as a function of degree, $n_{k,0}$, among numerous other results. We also examine fluctuations and concentration of joint degree-leafdegree empirical counts $N_{k,\ell}$. Although our main findings pertain to the preferential attachment tree, the approach we present is highly generalizable and can characterize numerous existing models, in addition to facilitating the development of tractable new models. We further demonstrate the approach by analyzing $n_{k,\ell}$ in two other models: the random recursive tree, and a redirection-based model.

💡 Research Summary

The paper introduces a novel local descriptor for growing random trees, namely the joint degree–leaf‑degree distribution (n_{k,\ell}), which records for each vertex both its ordinary degree (k) (the number of neighbours) and its leaf‑degree (\ell) (the number of neighbours that are leaves). While the degree distribution ({n_k}) has been extensively studied for preferential‑attachment (PA) trees, the statistics of leaves and, in particular, the number of leaf neighbours have received almost no attention. By augmenting the degree with leaf‑degree information, the authors obtain a richer statistical picture of the network’s boundary (leaves) and interior (protected vertices).

Model and Definitions
A PA tree is built by adding vertices one by one; each new vertex attaches to an existing vertex with probability proportional to that vertex’s degree. For a vertex (j) the degree is (k_j) and the leaf‑degree is (\ell_j). The random count (N_{k,\ell}) denotes how many vertices have the pair ((k,\ell)). Under the usual extensivity and self‑averaging assumptions, the normalized limits (n_{k,\ell}= \lim_{N\to\infty} N_{k,\ell}/N) exist and satisfy the normalization (\sum_{k>\ell\ge0} n_{k,\ell}=1) together with moment constraints (\sum_{k,\ell} k n_{k,\ell}=2) (the average degree of a tree) and (\sum_{k,\ell} \ell n_{k,\ell}=n_1) (the leaf fraction).

Generating‑function Framework
The authors introduce the bivariate generating function
\