Leaves of preferential attachment trees

2026-02-01

AI Summary

This study systematically characterizes the joint statistical properties of node degree and leaf-degree (i.e., the number of leaf neighbors) in preferential attachment trees. By constructing a multivariate generating function for the joint degree–leaf-degree distribution and employing probabilistic asymptotic analysis together with combinatorial stochastic process techniques, the authors derive for the first time the full joint distribution $n_{k,\ell}$, the marginal leaf-degree distribution $m_\ell$, and the proportion of protected nodes $n_{k,0}$. The proposed framework is highly generalizable and extends to various network models, including random recursive trees. Both theoretical analysis and numerical experiments confirm its validity, revealing the probabilistic regularities and concentration phenomena of local structures in large-scale networks.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Probabilistic Circuits and Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• preferential attachment trees
• degree distribution
• leaf degree
• protected nodes
• empirical fluctuations

Innovation

• preferential attachment
• leafdegree
• joint degree-leafdegree distribution
• multivariate generating function
• random recursive tree