Non-Uniform Distribution of Nodes in the Spatial Preferential Attachment Model

TL;DR

This paper investigates the spatial preferential attachment model with non-uniform node distribution, providing theoretical insights into network properties and methods for estimating node distances and regional densities.

Contribution

It extends the SPA model to non-uniform distributions, deriving precise theoretical results and practical estimators for network analysis.

Findings

The degree of a node depends on regional density.

Number of common neighbors varies with spatial proximity.

Estimators can reliably determine node distances and regional densities.

Abstract

The spatial preferential attachment (SPA) is a model for complex networks. In the SPA model, nodes are embedded in a metric space, and each node has a sphere of influence whose size increases if the node gains an in-link, and otherwise decreases with time. In this paper, we study the behaviour of the SPA model when the distribution of the nodes is non-uniform. Specifically, the space is divided into dense and sparse regions, where it is assumed that the dense regions correspond to coherent communities. We prove precise theoretical results regarding the degree of a node, the number of common neighbours, and the average out-degree in a region. Moreover, we show how these theoretically derived results about the graph properties of the model can be used to formulate a reliable estimator for the distance between certain pairs of nodes, and to estimate the density of the region containing a given node.