Leadership Statistics in Random Structures

TL;DR

This paper analytically investigates the universal statistical properties of leadership changes in evolving random structures like trees and graphs, revealing their self-similar nature and exponential decay in no-lead-change probability.

Contribution

It provides a unified analytical framework for understanding leadership statistics in random structures, highlighting universal behaviors and decay patterns.

Findings

Lead changes are rare and increase quadratically with log(system size).

Number of lead changes is self-similar over time.

Probability of no lead change decays exponentially.

Abstract

The largest component (``the leader'') in evolving random structures often exhibits universal statistical properties. This phenomenon is demonstrated analytically for two ubiquitous structures: random trees and random graphs. In both cases, lead changes are rare as the average number of lead changes increases quadratically with logarithm of the system size. As a function of time, the number of lead changes is self-similar. Additionally, the probability that no lead change ever occurs decays exponentially with the average number of lead changes.