On leaders and condensates in a growing network

TL;DR

This paper analyzes the Bianconi-Barabasi model of growing networks, revealing phase transitions, leader dynamics, and an infinite hierarchy of fluctuating condensates across different temperature regimes.

Contribution

It provides a detailed analysis of leader dynamics and introduces a novel picture of the condensed phase with fluctuating condensates.

Findings

Leaders appear endlessly on a doubly logarithmic time scale at finite temperature.

In the condensed phase, an infinite hierarchy of non-self-averaging condensates exists.

The model exhibits phase transitions between different growth regimes.

Abstract

The Bianconi-Barabasi model of a growing network is revisited. This model, defined by a preferential attachment rule involving both the degrees of the nodes and their intrinsic fitnesses, has the fundamental property to undergo a phase transition to a condensed phase below some finite critical temperature, for an appropriate choice of the distribution of fitnesses. At high temperature it exhibits a crossover to the Barabasi-Albert model, and at low temperature, where the fitness landscape becomes very rugged, a crossover to the recently introduced record-driven growth process. We first present an analysis of the history of leaders, the leader being defined as the node with largest degree at a given time. In the generic finite-temperature regime, new leaders appear endlessly, albeit on a doubly logarithmic time scale, i.e., extremely slowly. We then give a novel picture for the dynamics in the condensed phase. The latter is characterized by an infinite hierarchy of condensates, whose sizes are non-self-averaging and keep fluctuating forever.