Dynamical fitness models: evidence of universality classes for preferential attachment graphs

TL;DR

This paper introduces a family of preferential attachment models with fitness dynamics, revealing universality classes and phenomena like Bose-Einstein condensation in network growth.

Contribution

It defines a new class of fitness-based preferential attachment models with moving average fitness processes, connecting them to known models and exploring their phase behaviors.

Findings

Reproduction of features of Bianconi-Barabási and Barabási-Albert models

Identification of two regimes with distinct network properties

Discussion of Bose-Einstein condensation in the network context

Abstract

In this paper we define a family of preferential attachment models for random graphs with fitness in the following way: independently for each node, at each time step a random fitness is drawn according to the position of a moving average process with positive increments. We will define two regimes in which our graph reproduces some features of two well known preferential attachment models: the Bianconi-Barab\'asi and the Barab\'asi-Albert models. We will discuss a few conjectures on these models, including the convergence of the degree sequence and the appearance of Bose-Einstein condensation in the network when the drift of the fitness process has order comparable to the graph size.