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 birth times. 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.

Appendix A: Simulation of the model

In this section, we give an overview of the code used to generate the pictures shown throughout the paper. It is also used for estimating the limiting average clustering coefficient in Sect. 5. The code can be freely accessed at: http://www.mi.uni-koeln.de/~moerters/LoadPapers/adrc-model.R.

The main objective of the code is to sample neighbours of a given vertex (x, u) in the age-dependent random connection model in dimension 1 for given parameters $\beta$ and $\gamma$ and the profile function $\varphi$. Due to Proposition 4.1, which gives an explicit description of the neighbourhood of a given vertex, we can use rejection sampling to achieve this. The distribution in (5.1), defined on ${\mathbb {R}} \times (0,1]$, that we use to sample the neighbours of (x, u) may be unbounded and heavy tailed in the first parameter. To deal with this, we restrict the sampling to a region with mass $q = 0.99$ with respect to this distribution. This sampling works for arbitrary but reasonable choices of the profile function $\varphi$ and parameters $\beta$, $\gamma$; we provide and use an optimised sampling algorithm for with $a \ge \frac{1}{2}$. The advantage of studying this class of $\varphi$ is that expressions can be analytically simplified, which allows us to improve the algorithm by dividing the region from which the points are sampled into sub-regions with equal mass with respect to $\varphi$, thus increasing the acceptance rate for points sampled far away from (x, u). That is, the code first selects one of these equally likely sub-regions uniformly at random and then points are sampled therein until one is accepted. The numerical optimisation method nlminb is used to calculate the boundaries of the ranges, i.e., quantiles of the distribution from (5.1).

A first application of the sampling is the estimation of the expected local clustering coefficient of a vertex (0, u) in the age-dependent random connection model (see Fig. 3) and by Theorem 5.1 also the average clustering coefficient for the age-based preferential attachment network (see Fig. 4). To this end, the code samples pairs of neighbours of (0, u) and averages the probability that the pair is connected. A second application of the sampling is generating heatmaps of the neighbourhoods of a given vertex (see Fig. 2). The heatmaps are generated using the R library MASS and function kde2d by estimating the heat kernel for the sampled neighbouring vertices. Further properties thereof can be studied with additional heatmap generating functions that we provide.