Small World Model based on a Sphere Homeomorphic Geometry

TL;DR

This paper introduces a small world network model on a spherical surface, demonstrating efficient routing, low cycle formation probability, and a quadratic number of triangles, with implications for geometric network design.

Contribution

It presents a novel small world model based on sphere geometry, analyzing its routing efficiency and cycle properties, which advances understanding of geometric network structures.

Findings

Routing paths are expected to be logarithmic squared in size.

Probability of 3-cycle formation decreases inversely with size.

Number of triangles in the network scales quadratically with size.

Abstract

We define a small world model over the octahedron surface and relate its distances with those of embedded spheres, preserving constant bounded distortions. The model builds networks with both number of vertices and size $\Theta\left(n^2\right)$, where $n$ is the size parameter. It generates long-range edges with probability proportional to the inverse square of the distance between the vertices. We show a greedy routing algorithm that finds paths in the small world network with $\mathcal{O}\left(\log^2n\right)$ expected size. The probability of creating cycles of size three (C3) with long-range edges in a vertex is $\mathcal{O}\left(\log^{-1}n\right)$. Furthermore, there are $\Theta\left(n^2\right)$ expected number of C3's in the entire network.