The age-dependent random connection model

Abstract

We investigate a class of growing graphs embedded into the $d$-dimensional torus where new vertices arrive according to a Poisson process in time, are randomly placed in space and connect to existing vertices with a probability depending on time, their spatial distance and their relative ages. This simple model for a scale-free network is called the age-based spatial preferential attachment network and is based on the idea of preferential attachment with spatially induced clustering. We show that the graphs converge weakly locally to a variant of the random connection model, which we call the age-dependent random connection model. This is a natural infinite graph on a Poisson point process where points are marked by a uniformly distributed age and connected with a probability depending on their spatial distance and both ages. We use the limiting structure to investigate asymptotic degree distribution, clustering coefficients and typical edge lengths in the age-based spatial preferential attachment network.

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• Cluster-size decay in supercritical long-range percolation
• Distance evolutions in growing preferential attachment graphs
• Finiteness of the percolation threshold for inhomogeneous long-range models in one dimension
• Nearest-neighbour directed random hyperbolic graphs
• Chemical distance in geometric random graphs with long edges and scale-free degree distribution
• The contact process on scale-free geometric random graphs