Leaves of preferential attachment trees

TL;DR

The paper introduces the joint degree-leafdegree distribution $n_{k,\ell}$ as a rich descriptor of PA trees, deriving a closed two-index recursion and a generating-function PDE for $g(y,z)$ that encodes all $n_{k,\ell}$. From $n_{k,\ell}$, it obtains the leafdegree distribution $m_\ell$, the protected fraction $p$, and degree-stratified statistics, including a heavy-tailed $m_\ell$ with $m_\ell \sim \ell^{-3}$ and an exact $n_{k,0}$ form. It also analyzes fluctuations and self-averaging, proving asymptotic Gaussianity for fixed $(k,\ell)$ counts and presenting explicit results for $N_1$, $N_2$, and $N_{2,1}$, with covariances that define a multivariate normal limit. The approach is shown to be generalizable beyond PA trees, with detailed analyses for the random recursive tree and a redirection-based model in appendices, highlighting the framework's broad applicability to sparse graphs and related growth processes. Overall, the work provides a tractable, high-resolution description of leaf statistics in growing trees and opens avenues for richer model-building and empirical testing in complex networks.

Abstract

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.

Figures (5)

• Figure 1: Leafdegree distribution $M_\ell/N$ of a large PA tree ($N=10^8$), alongside theoretical values $m_\ell$ (Eq. \ref{['eq:ml_PA']}). The scattered semi-transparent dots represent the actual value of $M_{\ell}/N$, and the purple dotted line represents a log-binning thereof; the blue line is $\ell^{-3}$. Inset: comparison of $M_{\ell}/N$ (squares) and $m_{\ell}$ (dots) at small values of $\ell$.
• Figure 2: The fraction $N_{k,0}/N$ of vertices with degree $k$ and leafdegree $0$ (pink dots) in a large PA tree ($N=10^8$), in comparison with Eq. \ref{['eq:nk0']} for $n_{k,0}$ (green squares). Also depicted at $k=1$ is the analytical value of $p$ (black square) and measured protected fraction $P/N$ (magenta dot).
• Figure 3: The four types of transitions relevant to $N_{2,1}$, exemplified by the four structurally distinct attachment options labeled A, B, C, and D, in the $5$-vertex tree depicted. Gray vertices with dashed-line edges represent the distinct attachment possibilities. (Attachment to either of the two vertices labeled $D$ yields the same structural outcome.) The panels labeled (A), (B), (C), and (D) display the resulting tree for each possible attachment. From left to right: (A) depicts attachment to the leaf-neighbor of one of the $N_{2,1}$ vertices $j$ with $(k_j,\ell_j)=(2,1)$, (B) depicts direct attachment to one of those $N_{2,1}$ vertices, (C) depicts nonleaf attachment other than to one of the $N_{2,1}$, and (D) depicts attachment to a leaf that neighbors a nonleaf $j$ with $(k_j,\ell_j)\ne (2,1)$. Colors designate $(k_i,\ell_i)$ as depicted in the lower left box: black for $(1,0)$, orange for $(2,1)$, blue for $(2,0)$, green for $(3,2)$, cyan for $(3,1)$, and red for $(4,3)$. The probability weight in the PA tree for each of these possibilities is depicted in the lower right box; the actual probabilities for the PA tree are obtained by dividing by the total degree $\sum_{j}k_j=\sum_{k\ge 1}kN_{k}=2(N-1)$.
• Figure 4: A random tree from the CR model with $\alpha=-\infty$, $r=\frac{1}{2}$. Left: the full-sized tree with $N=2000$ vertices; colors distinguish leaves (green, $N_1=1016$), leafdegree-$1$ vertices (orange, $M_1=394$), and protected vertices (blue, $P=386$). Right: the subgraph consisting of all nonleaves, with the same color assignment.
• Figure 5: Leafdegree and degree distributions of the CR model at $\alpha=-\infty$ and $r=\frac{1}{2}$ as considered in Ref. QPA. The exact $m_\ell$ (blue squares), the numerical fractions $M_\ell/N$ in a single random tree of size $N=10^7$ (cyan circles), and the WKB asymptotic Eq. \ref{['ml_cr']} (red curve). The roughly estimated coefficient $C\approx 3.2$ is not analytically determined by the WKB approach. Additionally shown: the exact degree distribution $n_k$ (black squares) and numerical fractions $N_k/N$ (beige circles), and the exact and numerical protected fractions, $p$ and $P/N$ (dark green square and light green circle, respectively).