Condensation in preferential attachment models with location-based choice (2020)

Abstract: We introduce a new model of a preferential attachment based random graph which extends the family of models in which condensation phenomena can occur. Each vertex in our model has an associated uniform random variable which we refer to as its location. Our model evolves in discrete time by selecting r vertices from the graph with replacement, with sampling probabilities proportional to their degrees plus a constant α. A new vertex joins the network and attaches to one of these r vertices according to a given probability associated to the ranking of their locations. Using stochastic approximation techniques we give conditions for the occurrence of condensation in this model, showing the existence of phase transitions in α below which condensation occurs. The condensation in our model differs from that in preferential attachment models with fitness in that the condensation can occur at a random location, that it can (but not necessarily) be due to a persistent hub, and that there can be more than one point of condensation.

The pictures (produced by John Haslegrave) below show simulations illustrating two different types of condensation in the case where the new vertex picks a sample of three and connects to the middle one. In the first picture, a persistent hub appears to emerge rapidly, while in the second picture slower convergence appears to be taking place with no persistent hub emerging. The parameters are the same in each case; for more details see section 4.1 of the paper.