Clustering Properties of Spatial Preferential Attachment Model

In this paper, we study the clustering properties of the Spatial Preferential Attachment (SPA) model introduced by Aiello et al. in 2009. This model naturally combines geometry and preferential attachment using the notion of spheres of influence. It was previously shown in several research papers that graphs generated by the SPA model are similar to real-world networks in many aspects. For example, the vertex degree distribution was shown to follow a power law. In the current paper, we study the behaviour of C(d), which is the average local clustering coefficient for the vertices of degree d. This characteristic was not previously analyzed in the SPA model. However, it was empirically shown that in real-world networks C(d) usually decreases as \(d^{-a}\) for some \(a>0\) and it was often observed that \(a=1\). We prove that in the SPA model C(d) decreases as \(1{\slash }d\). Furthermore, we are also able to prove that not only the average but the individual local clustering coefficient of a vertex v of degree d behaves as \(1{\slash }d\) if d is large enough. The obtained results are illustrated by numerous experiments with simulated graphs.