Spectral embedding of weighted graphs

TL;DR

This paper investigates how transforming edge weights in spectral embedding of weighted graphs can improve community detection, providing theoretical insights and practical recommendations for various types of networks.

Contribution

It formalizes the impact of weight transformations on spectral embedding quality and offers guidance on effective transformations like tempering or thresholding.

Findings

Transformations can significantly enhance community distinguishability.

Theoretical analysis supports practical benefits of weight transformations.

Different network types benefit from specific weight adjustment strategies.

Abstract

When analyzing weighted networks using spectral embedding, a judicious transformation of the edge weights may produce better results. To formalize this idea, we consider the asymptotic behavior of spectral embedding for different edge-weight representations, under a generic low rank model. We measure the quality of different embeddings -- which can be on entirely different scales -- by how easy it is to distinguish communities, in an information-theoretic sense. For common types of weighted graphs, such as count networks or p-value networks, we find that transformations such as tempering or thresholding can be highly beneficial, both in theory and in practice.