Extensive Condensation in a model of Preferential Attachment with Fitnesses

TL;DR

This paper introduces a new preferential attachment model with fitness, explaining the condensation phenomenon where highly fit vertices dominate, using a duality with branching-coalescing particles, and shows this condensation is extensive in large graphs.

Contribution

It establishes a duality framework for preferential attachment with fitness and explains the extensive condensation phenomenon within this model.

Findings

Condensation arises from a growth-extinction dichotomy in the dual process.

Unusually fit vertices become neighbors to a non-vanishing proportion of the graph.

Condensation is extensive and persists as the graph grows.

Abstract

We introduce a new model of preferential attachment with fitness, and establish a time reversed duality between the model and a system of branching-coalescing particles. Using this duality, we give a clear and concise explanation for the condensation phenomenon, in which unusually fit vertices may obtain abnormally high degree: it arises from a growth-extinction dichotomy within the branching part of the dual. We show further that the condensation is extensive. As the graph grows, unusually fit vertices become, each only for a limited time, neighbouring to a non-vanishing proportion of the current graph.