A spatial preferential attachment model with local clustering

Abstract

A class of growing networks is introduced in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favouring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Most notably, empirical degree distributions converge to a limit law, which can be a power law with any exponent τ > 2, and the average clustering coefficient converges to a positive limit. Our main tool to show these and other results is a weak law of large numbers in the spirit of Penrose and Yukich, which can be applied thanks to a novel rescaling idea. We also conjecture that the networks have a robust giant component if τ is sufficiently small. © 2013 Springer International Publishing.