Identifying the latent space geometry of network models through analysis of curvature

TL;DR

This paper introduces a method to identify the underlying geometric structure of network models by estimating manifold type, dimension, and curvature from network data, aiding understanding of community formation and repulsion.

Contribution

It develops hypothesis tests for determining if network distances can be embedded in specific constant curvature geometries, advancing network modeling techniques.

Findings

Successfully estimates manifold curvature and dimension from network data.

Provides hypothesis tests to distinguish between different geometric embeddings.

Applied to economics and neuroscience datasets, demonstrating practical utility.

Abstract

A common approach to modeling networks assigns each node to a position on a low-dimensional manifold where distance is inversely proportional to connection likelihood. More positive manifold curvature encourages more and tighter communities; negative curvature induces repulsion. We consistently estimate manifold type, dimension, and curvature from simply connected, complete Riemannian manifolds of constant curvature. We represent the graph as a noisy distance matrix based on the ties between cliques, then develop hypothesis tests to determine whether the observed distances could plausibly be embedded isometrically in each of the candidate geometries. We apply our approach to data-sets from economics and neuroscience.