Statistics of Lead Changes in Popularity-Driven Systems

TL;DR

This paper analyzes how the most popular nodes in growing networks change over time, revealing that lead changes grow logarithmically with network size and depend on the growth mechanism.

Contribution

It provides a statistical analysis of lead change dynamics in popularity-driven networks, highlighting the impact of growth mechanisms on lead retention probabilities.

Findings

Number of lead changes increases logarithmically with network size.

Probability of initial leader retaining lead approaches a constant in popularity-driven growth.

Probability decays polynomially with network size in unbiased growth.

Abstract

We study statistical properties of the highest degree, or most popular, nodes in growing networks. We show that the number of lead changes increases logarithmically with network size N, independent of the details of the growth mechanism. The probability that the first node retains the lead approaches a finite constant for popularity-driven growth, and decays as N^{-phi}(ln N)^{-1/2}, with phi=0.08607..., for growth with no popularity bias.