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Random friend trees

Louigi Addario Berry, Simon Briend, Luc Devroye, Serte Donderwinkel, Céline Kerriou, Gábor Lugosi

TL;DR

A random recursive tree model featuring complete redirection called the random friend tree, introduced by Saramaki and Kaski, and proven on the neighbourhood of fixed vertices and edges is studied, providing insights into the overall structure of the tree.

Abstract

We study a random recursive tree model featuring complete redirection called the random friend tree and introduced by Saramäki and Kaski. Vertices are attached in a sequential manner one by one by selecting an existing target vertex and connecting to one of its neighbours (or friends), chosen uniformly at random. This model has interesting emergent properties, such as a highly skewed degree sequence. In contrast to the preferential attachment model, these emergent phenomena stem from a local rather than a global attachment mechanism. The structure of the resulting tree is also strikingly different from both the preferential attachment tree and the uniform random recursive tree: every edge is incident to a macro-hub of asymptotically linear degree, and with high probability all but at most $n^{9/10}$ vertices in a tree of size $n$ are leaves. We prove various results on the neighbourhood of fixed vertices and edges, and we study macroscopic properties such as the diameter and the degree distribution, providing insights into the overall structure of the tree. We also present a number of open questions on this model and related models.

Random friend trees

TL;DR

A random recursive tree model featuring complete redirection called the random friend tree, introduced by Saramaki and Kaski, and proven on the neighbourhood of fixed vertices and edges is studied, providing insights into the overall structure of the tree.

Abstract

We study a random recursive tree model featuring complete redirection called the random friend tree and introduced by Saramäki and Kaski. Vertices are attached in a sequential manner one by one by selecting an existing target vertex and connecting to one of its neighbours (or friends), chosen uniformly at random. This model has interesting emergent properties, such as a highly skewed degree sequence. In contrast to the preferential attachment model, these emergent phenomena stem from a local rather than a global attachment mechanism. The structure of the resulting tree is also strikingly different from both the preferential attachment tree and the uniform random recursive tree: every edge is incident to a macro-hub of asymptotically linear degree, and with high probability all but at most vertices in a tree of size are leaves. We prove various results on the neighbourhood of fixed vertices and edges, and we study macroscopic properties such as the diameter and the degree distribution, providing insights into the overall structure of the tree. We also present a number of open questions on this model and related models.
Paper Structure (16 sections, 37 theorems, 170 equations, 4 figures)

This paper contains 16 sections, 37 theorems, 170 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Local results
  4. Global results
  5. Proofs of local properties
  6. Hubs
  7. Expected degree of $W_n$
  8. Eternal leaves and eternal degree k vertices
  9. Proofs of global properties
  10. Typical distances
  11. Diameter
  12. Leaf-depth
  13. High-degree vertices
  14. Low-degree vertices
  15. Open questions and future directions
  16. ...and 1 more sections

Key Result

Theorem 3.1

For vertex $u\in\mathbb{N}$, the random variables $D_n(u)$, $L_n(u)$ and $Z_u$, defined in Section sec:Notations, are such that almost surely as $n\to \infty$.

Figures (4)

  • Figure 1: A realisation up to $n=9$, with $V_n$ in blue and $W_n$ in yellow.
  • Figure 2: A realisation of $T_n$ with $n=1000$.
  • Figure 3: Illustration of a RFT of size $25$ and diameter $6$, with vertices of one of the paths of length $6$ highlighted.
  • Figure 4: Illustration of a coupling between $B$ (in blue) and $S$ (in red).

Theorems & Definitions (64)

  • Theorem 3.1: Convergence of normalised degree
  • Theorem 3.2: Abundance of hubs
  • Theorem 3.3: Expected degree of $W_n$
  • Theorem 3.4: Bounded degree
  • Theorem 4.1: Typical distances
  • Theorem 4.2: Diameter of random friend trees
  • Theorem 4.3: Leaf depth
  • Theorem 4.4: Leaf depth in URRT
  • Theorem 4.5: Number of hubs
  • Theorem 4.6: Abundance of high degree vertices
  • ...and 54 more