Structural Properties of the Geometric Preferential Attachment Model

TL;DR

This paper investigates the structural properties of networks generated by the geometric preferential attachment model, revealing similarities and differences with standard models in terms of triangles, degree growth, connectivity, and diameter.

Contribution

It provides a rigorous analysis of key network properties under geometric constraints, extending existing models to include spatial factors and finite out-degrees.

Findings

Expected number of triangles proportional to standard model

Maximum degree grows polynomially with network size

Connectivity and diameter results extended to finite out-degree networks

Abstract

This paper analyzes key properties of networks generated by geometric preferential attachment. We establish that the expected number of triangles is proportional to that of the standard preferential attachment model, with a proportionality constant equal to the ratio of the number of triangles between a random geometric graph and an Erd\H{o}s-R\'enyi graph. Furthermore, we prove that the maximum degree grows polynomially with the network size, sharing the same exponent as the standard model; however, the spatial constraint induces a slower growth rate in the network's early evolution. Finally, we extend prior results on connectivity and diameter to the case of networks with finite out-degrees.