A Spatial Preferential Attachment Model with Local Clustering
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.
KeywordsScale-free network Barabasi-Albert model preferential attachment dynamical random graph geometric random graph power law degree distribution edge length distribution clustering coefficient
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