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Structural results for the Tree Builder Random Walk

Janos Engländer, Giulio Iacobelli, Gábor Pete, Rodrigo Ribeiro

Abstract

We study the Tree Builder Random Walk: a randomly growing tree, built by a walker as she is walking around the tree. Namely, at each time $n$, she adds a leaf to her current vertex with probability $p_n \asymp n^{-γ}$, $γ\in (2/3,1]$, then moves to a uniform random neighbor on the possibly modified tree. We show that the tree process at its growth times, after a random finite number of steps, can be coupled to be identical to the Barabási-Albert preferential attachment tree model. Thus, our TBRW-model is a local dynamics giving rise to the BA-model. The coupling also implies that many properties known for the BA-model, such as diameter and degree distribution, can be directly transferred to our TBRW-model, extending previous results.

Structural results for the Tree Builder Random Walk

Abstract

We study the Tree Builder Random Walk: a randomly growing tree, built by a walker as she is walking around the tree. Namely, at each time , she adds a leaf to her current vertex with probability , , then moves to a uniform random neighbor on the possibly modified tree. We show that the tree process at its growth times, after a random finite number of steps, can be coupled to be identical to the Barabási-Albert preferential attachment tree model. Thus, our TBRW-model is a local dynamics giving rise to the BA-model. The coupling also implies that many properties known for the BA-model, such as diameter and degree distribution, can be directly transferred to our TBRW-model, extending previous results.
Paper Structure (23 sections, 25 theorems, 151 equations, 1 figure)

This paper contains 23 sections, 25 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The TBRW model
  3. The Barabási-Albert model
  4. Applications
  5. Chain Walk Catalyst
  6. On the ubiquity of linear preferential attachment and an approach to it via TBRW
  7. Physical networks.
  8. Main results
  9. Proof ideas and technical difficulties
  10. Organization of the paper
  11. Preliminaries
  12. Mixing time upper bound
  13. Strong Stationary Times
  14. Strong Stationary times for the TBRW
  15. Proof of Theorem \ref{['thm:generaltransfer']}: General Condition for Transfer Principle
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\mathcal{G}$ be an asymptotic graph property. Consider a TBRW satisfying condition (M) of Definition def:TP. Then,

Figures (1)

  • Figure 1: Simulation of TBRW with 1000 vertices, with different values of $\gamma$. (Thanks to Á. Kúsz for the pictures.)

Theorems & Definitions (61)

  • Definition 1.1: Asymptotic Graph Property
  • Remark 1.1
  • Example 1.1: Linear growth
  • Example 1.2: The height of the BA-tree
  • Example 1.3: The maximum degree of the BA-tree
  • Definition 1.2: Condition (M)
  • Theorem 1.1: Transfer Principle under (M)
  • Theorem 1.2: A family of TBRW satisfying (M)
  • Corollary 1.1: Power law for degree distribution
  • Corollary 1.2: Height of $T_n$
  • ...and 51 more