The containment profile of hyperrecursive trees

TL;DR

This paper analyzes the evolution of vertex containment levels in a hypergraph grown recursively, providing exact distributions and asymptotic behaviors for local and global profiles, revealing phase transitions and normality in large-scale limits.

Contribution

It introduces a detailed analysis of containment profiles in hyperrecursive trees, deriving exact and asymptotic distributions, and identifying phase transitions in vertex containment levels.

Findings

Exact mean, variance, and distribution for local containment profiles.

Asymptotic normality for early and global containment levels.

Phase transitions in distribution types across vertex positions.

Abstract

We investigate vertex levels of containment in a random hypergraph grown in the spirit of a recursive tree. We consider a local profile tracking the evolution of the containment of a particular vertex over time, and a global profile concerned about counts of the number of vertices of a particular containment level. For the local containment profile, we obtain the exact mean, variance and probability distribution in terms of standard combinatorial quantities like generalized harmonic numbers and Stirling numbers of the first kind. Asymptotically, we observe phases: the early vertices have an asymptotically normal distribution, intermediate vertices have a Poisson distribution, and late vertices have a degenerate distribution. As for the global containment profile, we establish an asymptotically normal distribution for the number of vertices at the smallest containment level as well as their covariances with the number of vertices at the second smallest containment level and the variances of these numbers.