Spatial preferential attachment networks: Power laws and clustering coefficients

TL;DR

This paper introduces a spatial preferential attachment network model where new nodes connect based on proximity and degree, resulting in power-law degree distributions, positive clustering, and phase transitions at critical exponents.

Contribution

It presents a novel spatial network model combining preferential attachment with spatial clustering, analyzing its degree distribution and clustering properties.

Findings

Degree distributions can follow any power law with exponent > 2

Average clustering coefficient converges to a positive limit

Phase transition in clustering occurs at exponent τ=3

Abstract

We define a class of growing networks in which new nodes are given a spatial position and are connected to existing nodes with a probability mechanism favoring short distances and high degrees. The competition of preferential attachment and spatial clustering gives this model a range of interesting properties. Empirical degree distributions converge to a limit law, which can be a power law with any exponent $\tau>2$. The average clustering coefficient of the networks converges to a positive limit. Finally, a phase transition occurs in the global clustering coefficients and empirical distribution of edge lengths when the power-law exponent crosses the critical value $\tau=3$. Our main tool in the proof of these results is a general weak law of large numbers in the spirit of Penrose and Yukich.