Extensive Condensation in a model of Preferential Attachment with Fitnesses

Nic Freeman, Jonathan Jordan

Published 2018-12-17Version 1

We introduce a new model of preferential attachment with fitness, and establish a time reversed duality between our 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 an explosion-extinction dichotomy within the branching part of the dual. We show further that, in our model, 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.