A Note on Markov Normalized Magnetic Eigenmaps

TL;DR

This paper proposes using the Markov transition matrix to construct the magnetic Laplacian in magnetic eigenmaps, resulting in more stable embeddings and convergence to page rank, with empirical improvements demonstrated.

Contribution

It introduces a normalization technique for magnetic Laplacians that enhances stability and convergence properties in magnetic eigenmaps.

Findings

Embedding stability improves with Markov normalization

Eigenvector convergence to page rank is achieved

Empirical results show improved embeddings

Abstract

We note that building a magnetic Laplacian from the Markov transition matrix, rather than the graph adjacency matrix, yields several benefits for the magnetic eigenmaps algorithm. The two largest benefits are that the embedding becomes more stable as a function of the rotation parameter g, and the principal eigenvector of the magnetic Laplacian now converges to the page rank of the network as a function of diffusion time. We show empirically that this normalization improves the phase and real/imaginary embeddings of the low-frequency eigenvectors of the magnetic Laplacian.