Local clustering coefficient of spatial preferential attachment model
In this article, we study the clustering properties of the spatial preferential attachment (SPA) model. 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. Also, this model was successfully used for several practical applications. However, the clustering properties of the SPA model were not fully analysed. The clustering coefficient is an important characteristic of complex networks which is tightly connected with its community structure. In this article, we study the behaviour of C(d)C(d), which is the average local clustering coefficient for the vertices of degree dd. It was empirically shown that in real-world networks C(d)C(d) usually decreases as d−ad−a for some a>0a>0 and it was often observed that a=1a=1. We prove that in the SPA model C(d)C(d) decreases as 1/d1/d. Furthermore, we are also able to prove that not only the average but also the individual local clustering coefficient of a vertex vv of degree dd behaves as 1/d1/d if dd is large enough. The obtained results further confirm the suitability of the SPA model for fitting various real-world complex networks.