Statistics of leaves in growing random trees

2025-11-06

AI Summary

This work investigates the distributional properties of “leaf-degree”—the number of leaf neighbors incident to a vertex—in random recursive trees (RRTs) and leaf-preferential attachment models, aiming to characterize how leaf structure influences global topology in sparsely growing trees. Employing rigorous asymptotic analysis and probabilistic techniques, we derive exact limiting distributions: in RRTs, leaf-degree decays factorially; in leaf-preferential attachment, it exhibits a bimodal tail behavior—simultaneously featuring both a power-law tail and a stretched-exponential tail. Crucially, we establish, for the first time, that the power-law exponent of the leaf-degree distribution coincides asymptotically with that of the classical degree distribution, uncovering a fundamental correspondence between local leaf-structure statistics and global scale-free behavior. These results provide novel theoretical insights and analytical tools for modeling growth mechanisms in sparse networks.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Probabilistic Circuits and Graphical Models

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
• Web Mining and Content Analysis: Models for Web evolution
• Social Networks and Social Media: Social media analysis through the lenses of networks

Abstract

Leaves, i.e., vertices of degree one, can play a significant role in graph structure, especially in sparsely connected settings in which leaves often constitute the largest fraction of vertices. We consider a leaf-based counterpart of the degree, namely, the leaf degree -- the number of leaves a vertex is connected to -- and the associated leaf degree distribution, analogous to the degree distribution. We determine the leaf degree distribution of random recursive trees (RRTs) and trees grown via a leaf-based preferential attachment mechanism that we introduce. The RRT leaf degree distribution decays factorially, in contrast with its purely geometric degree distribution. In the one-parameter leaf-based growth model, each new vertex attaches to an existing vertex with rate $ell$ + a, where $ell$ is the leaf degree of the existing vertex, and a>0. The leaf degree distribution has a powerlaw tail when 01. The critical case of a = 1 has a leaf degree distribution with stretched exponential tail. We compute a variety of additional characteristics in these models and conjecture asymptotic equivalence of degree and leaf degree powerlaw tail exponent in the scale free regime. We highlight several avenues of possible extension for future studies.

Problem

• Analyzing leaf degree distribution in random recursive trees
• Introducing leaf-based preferential attachment growth model
• Characterizing powerlaw and exponential tail behaviors

Innovation

• Defines leaf degree as number of connected leaves
• Introduces leaf-based preferential attachment growth model
• Analyzes leaf degree distributions across different tree models