Solvable Metric Growing Networks

TL;DR

This paper introduces a model for growing networks embedded in metric spaces, where connection probabilities depend on geographic distances, and analyzes key properties like degree distribution and shortest paths.

Contribution

It proposes a novel network growth model incorporating geographic distance effects and provides analytical insights into its structural properties.

Findings

Mean degree depends on spatial distribution

Degree distribution follows a specific pattern influenced by geography

Shortest path lengths are affected by the spatial embedding

Abstract

Structure and dynamics of complex networks usually deal with degree distributions, clustering, shortest path lengths and other graph properties. Although these concepts have been analysed for graphs on abstract spaces, many networks happen to be embedded in a metric arrangement, where the geographic distance between vertices plays a crucial role. The present work proposes a model for growing network that takes into account the geographic distance between vertices: the probability that they are connected is higher if they are located nearer than farther. In this framework, the mean degree of vertices, degree distribution and shortest path length between two randomly chosen vertices are analysed.