Local Clustering Coefficient of Spatial Preferential Attachment Model

TL;DR

This paper analyzes the clustering properties of the Spatial Preferential Attachment (SPA) model, demonstrating that the local clustering coefficient decreases as 1/d, aligning with real-world network observations.

Contribution

It provides the first rigorous proof that the local clustering coefficient in the SPA model decreases as 1/d, matching empirical patterns in real networks.

Findings

C(d) decreases as 1/d in the SPA model

Individual local clustering coefficient behaves as 1/d for large degree d

Results support SPA model's applicability to real-world networks

Abstract

In this paper, 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 analyzed. The clustering coefficient is an important characteristic of complex networks which is tightly connected with its community structure. In the current paper, we study the behaviour of C(d), which is the average local clustering coefficient for the vertices of degree d. It was empirically shown that in real-world networks C(d) usually decreases as 1/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/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/d if d is large enough. The obtained results further confirm the suitability of the SPA model for fitting various real-world complex networks.