[Paper Review] On asymptotics of two non-uniform recursive tree models

Ella Hiesmayr|arXiv (Cornell University)|Oct 3, 2017

Stochastic processes and statistical mechanics14 references3 citations

TL;DR

This thesis analyzes two non-uniform recursive tree models: weighted recursive trees (WRT), where node attachment probabilities depend on node-specific weights, and biased recursive trees (BRT), constructed using riffle shuffle permutations instead of uniform random permutations. It shows that WRT asymptotically resemble uniform recursive trees (URT) under broad conditions, while BRT exhibit distinct behavior that converges to URT depending on riffle shuffle parameters, with key results on depth, leaf counts, and branch numbers established via coupling and martingale methods.

ABSTRACT

In this thesis the properties of two kinds of non-uniform random recursive trees are studied. In the first model weights are assigned to each node, thus altering the attachment probabilities. We will call these trees weighted recursive trees. In the second model a different distribution rather than the uniform one is chosen on the symmetric group, namely a riffle shuffle distribution. These trees will be called biased recursive trees. For both of these models the number of branches, the number of leaves, the depth of nodes and some other properties are studied. The focus is on asymptotic results and the comparison with uniform random recursive trees. It will be shown that the studied properties of weighted recursive trees are close to uniform recursive trees in many cases when the number of nodes increases. In contrast biased recursive trees show a different behaviour but approach uniform recursive trees depending on the parameters of the riffle shuffle distribution.

Motivation & Objective

• To understand the asymptotic behavior of non-uniform recursive tree models, particularly in comparison to uniform recursive trees (URT).
• To investigate how node weights and non-uniform attachment mechanisms—specifically riffle shuffle distributions—affect structural properties like depth, number of leaves, and number of branches.
• To establish conditions under which weighted recursive trees (WRT) asymptotically approach the behavior of URT, and to analyze the convergence of biased recursive trees (BRT) to URT under varying riffle shuffle parameters.
• To develop coupling and martingale techniques for transferring known results from URT to WRT and BRT, enabling asymptotic and concentration analysis.

Proposed method

• Uses a coupling construction between weighted recursive trees (WRT) and uniform recursive trees (URT) to compare asymptotic distributions of structural statistics such as depth and number of leaves.
• Applies martingale difference sequences and the Azéma-Yor embedding to analyze the depth of nodes in WRT, particularly under non-uniform weight sequences.
• Employs the theory of riffle shuffles and anti-records to model the attachment mechanism in biased recursive trees (BRT), leveraging independence structures in permutation statistics.
• Utilizes the multivariate normal approximation and central limit theorems for sums of dependent random variables in WRT, especially when weights are fixed or follow specific sequences.
• Applies the Wasserstein and Kolmogorov metrics to quantify convergence in distribution between WRT and URT, and between BRT and URT under parameter variation.
• Analyzes the number of branches in BRT using cycle structure of riffle shuffle permutations, though a full asymptotic distribution remains open.

Experimental results

Research questions

• RQ1 Under what conditions do weighted recursive trees (WRT) asymptotically approach the statistical behavior of uniform recursive trees (URT)?
• RQ2 How does the distribution of node depth in WRT behave as the number of nodes increases, particularly when weights deviate from uniformity?
• RQ3 What is the asymptotic distribution of the number of branches in biased recursive trees (BRT) constructed via riffle shuffle permutations?
• RQ4 How do the parameters of the riffle shuffle distribution affect the convergence of BRT to URT in terms of depth and leaf count?
• RQ5 Can coupling techniques be used to derive concentration inequalities for structural statistics in WRT and BRT based on known results for URT?

Key findings

• Weighted recursive trees (WRT) with finitely many non-unit weights exhibit asymptotic behavior close to uniform recursive trees (URT), particularly in the distribution of node depth and number of leaves.
• For WRT, the expected depth of a node converges to a value dependent on the weight sequence, and under certain conditions, the depth distribution satisfies a central limit theorem.
• Biased recursive trees (BRT) constructed from riffle shuffles with $a$ piles show that the maximum degree is exactly $a$, suggesting structural similarity to $a$-ary trees.
• The expectation of node depth in BRT depends asymptotically only on the first parameter $p_1$ of the riffle shuffle, indicating a strong influence of the initial distribution.
• The number of branches in BRT remains an open problem due to global dependence in anti-records, though cycle structure of riffle permutations offers a potential path forward.
• Coupling constructions between WRT and URT enable transfer of asymptotic and concentration results, suggesting broader applicability to other tree statistics.