Robust analysis of preferential attachment models with fitness

TL;DR

This paper introduces a robust method to analyze preferential attachment models with fitness, revealing phenomena like Bose-Einstein condensation and providing detailed degree and fitness distributions.

Contribution

A new, model-agnostic analytical approach for preferential attachment with fitness, extending previous work and uncovering Bose-Einstein condensation in various models.

Findings

Bose-Einstein condensation can occur in a wide range of models.

The method provides the fitness distribution of a typical vertex.

Joint degree and fitness distribution of a randomly chosen vertex was computed.

Abstract

The preferential attachment network with fitness is a dynamic random graph model. New vertices are introduced consecutively and a new vertex is attached to an old vertex with probability proportional to the degree of the old one multiplied by a random fitness. We concentrate on the typical behaviour of the graph by calculating the fitness distribution of a vertex chosen proportional to its degree. For a particular variant of the model, this analysis was first carried out by Borgs, Chayes, Daskalakis and Roch. However, we present a new method, which is robust in the sense that it does not depend on the exact specification of the attachment law. In particular, we show that a peculiar phenomenon, referred to as Bose-Einstein condensation, can be observed in a wide variety of models. Finally, we also compute the joint degree and fitness distribution of a uniformly chosen vertex.