Structural Properties of the Geometric Preferential Attachment Model
Chenxu Feng, Yifan Li
TL;DR
This work analyzes the geometric preferential attachment model (GPM), a synthesis of spatial locality and degree-biased attachment on a sphere, to capture clustering and community structure in networks. It derives precise asymptotics for key structural features: triangles scale with a geometry-enhanced factor leading to $T_n=(1+o(1))\frac{m(m-1)(1+\delta)^2(m(1+\delta)+1)F_p}{(2+\delta)\delta^2 p}\log n$, maximum degree grows as $\Theta\big(\log(p^{-1})^{(1+\delta)/(2+\delta)} (np)^{1/(2+\delta)}\big)$ (with a convergent limit after normalization), and a sharp connectivity threshold occurs near $n\sim p^{-m/(m-1)}$ with a consequent diameter of order $\log n$. The analysis hinges on concentration bounds for the candidate-set weight $L(n)$, subgraph probability estimates, and a second-moment method for triangles, all extended to a general preference function $f$ where applicable. The results demonstrate how spatial constraints amplify clustering while preserving PAM-like degree growth and small-world diameter, and they establish a framework for understanding connectivity transitions in geometric networks with constant out-degree. These findings have implications for modeling real-world networks where geometry and preferential attachment interact and for assessing the impact of spatial scales on network connectivity and resilience.
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 Erdős-Rényi 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.