A study of centrality measures in random recursive trees

TL;DR

This paper analyzes five classical centrality measures in uniform random recursive trees, focusing on the root's centrality and the timing of the most central vertex, with implications for network archaeology.

Contribution

It provides a detailed probabilistic analysis of centrality measures in recursive trees, including root centrality, rank, and persistence properties, which was previously unexplored.

Findings

Probability that the root is the most central vertex

Expected rank of the root under each measure

Size of the top-ranked vertex set containing the root

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.