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A study of centrality measures in random recursive trees

Richard Coll Josifov, Luc Devroye, Gabor Lugosi

Abstract

We investigate the behaviour of five classical centrality measures--Jordan, rumor, betweenness, degree, and closeness centralities--in the setting of uniform random recursive trees. Motivated by applications in network archaeology, we focus on two fundamental questions: (i) the birth index (time of arrival) of the most central vertex, and (ii) the relative centrality of the root. We quantify the probability that the root is the most central vertex, analyze its expected rank under each centrality measure, and determine the expected birth index of a central vertex. In addition, we characterize the typical size of the set of top-ranked vertices that contains the root with high probability. Finally, for each centrality notion, we study the persistence properties of the center and the asymptotic behaviour of the root's rank.

A study of centrality measures in random recursive trees

Abstract

We investigate the behaviour of five classical centrality measures--Jordan, rumor, betweenness, degree, and closeness centralities--in the setting of uniform random recursive trees. Motivated by applications in network archaeology, we focus on two fundamental questions: (i) the birth index (time of arrival) of the most central vertex, and (ii) the relative centrality of the root. We quantify the probability that the root is the most central vertex, analyze its expected rank under each centrality measure, and determine the expected birth index of a central vertex. In addition, we characterize the typical size of the set of top-ranked vertices that contains the root with high probability. Finally, for each centrality notion, we study the persistence properties of the center and the asymptotic behaviour of the root's rank.
Paper Structure (19 sections, 31 theorems, 94 equations, 2 figures, 2 tables)

This paper contains 19 sections, 31 theorems, 94 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Rank of the root and index of the center
  3. Related work
  4. Notation
  5. Results
  6. Preliminaries
  7. Jordan centrality
  8. Root estimation and expected rank of the root
  9. Persistence
  10. Betweenness centrality
  11. Probability that the root is the centroid
  12. Tightness, expected rank of the root and root estimation
  13. Persistence
  14. Degree centrality
  15. Rumor centrality
  16. ...and 4 more sections

Key Result

Lemma 2

Let $T_n$ be a tree on n vertices. Then Jordan centrality satisfies the following properties.

Figures (2)

  • Figure 1: Representation of the subtrees in the path from the root to vertex $v$.
  • Figure 2: Representation of the subtree of $v_{1^{(3)}}$ and paths of length $2$ within it.

Theorems & Definitions (56)

  • Remark 1
  • Remark 2
  • Definition 1
  • Remark 3
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • Lemma 5
  • proof
  • ...and 46 more