Characterization and comparison of large directed graphs through the spectra of the magnetic Laplacian

TL;DR

This paper explores using the magnetic Laplacian to analyze large directed graphs, revealing community structures and enabling scalable network comparisons through spectral methods and the Kernel Polynomial Method.

Contribution

It introduces a scalable spectral approach for directed graphs using the magnetic Laplacian, combining KPM and Wasserstein metrics, and provides an open-source Python package for large network analysis.

Findings

Community structure relates to spectral symmetry in stochastic block models.

The approach scales to networks with hundreds of thousands of nodes.

The method effectively measures distances between large directed networks.

Abstract

In this paper we investigated the possibility to use the magnetic Laplacian to characterize directed graphs (a.k.a. networks). Many interesting results are obtained, including the finding that community structure is related to rotational symmetry in the spectral measurements for a type of stochastic block model. Due the hermiticity property of the magnetic Laplacian we show here how to scale our approach to larger networks containing hundreds of thousands of nodes using the Kernel Polynomial Method (KPM). We also propose to combine the KPM with the Wasserstein metric in order to measure distances between networks even when these networks are directed, large and have different sizes, a hard problem which cannot be tackled by previous methods presented in the literature. In addition, our python package is publicly available at \href{https://github.com/stdogpkg/emate}{github.com/stdogpkg/emate}. The codes can run in both CPU and GPU and can estimate the spectral density and related trace functions, such as entropy and Estrada index, even in directed or undirected networks with million of nodes.