Magnetic Eigenmaps for the visualization of directed networks

Abstract

We propose a framework for the visualization of directed networks relying on the eigenfunctions of the magnetic Laplacian, called here Magnetic Eigenmaps. The magnetic Laplacian is a complex deformation of the well-known combinatorial Laplacian. Features such as density of links and directionality patterns are revealed by plotting the phases of the first magnetic eigenvectors. An interpretation of the magnetic eigenvectors is given in connection with the angular synchronization problem. Illustrations of our method are given for both artificial and real networks.

A note on Markov normalized magnetic eigenmaps

“…Proof. From Theorem 1 in [2], we know that the magnetic Laplacian L has a zero eigenvalue iff ∃h such that a i,j = h j − h i . Furthermore, we know that, in this case, φ…”

“…Three Cluster Example. We begin with the three cluster example from [2]. This example is particularly simple given that the complex rotation of three clusters will, more often than not, keep the clusters separated.…”

“…Another issue with not including a diffusion time is the need for g to implicitly assume the number of clusters. By setting g to be 1/k for k clusters as the authors suggest [2], the complex rotation can lose valuable information about transitions that aren't k periodic. We demonstrate this with an example similar to the three cluster example from Section 3.1.…”

“…We consider the problem of building a diffusion operator on directed networks and graphs with adjacency matrix W ∈ R N ×N that parallels Diffusion Maps on undirected networks [1]. A natural approach to this is to build the Magnetic Eigenmaps Laplacian [2],…”

Directed network Laplacians and random graph models

“…-We establish connections between these random graph models and algorithms from [6,11] that use the magnetic Laplacian and trophic Laplacian, respectively, by showing that reordering nodes or mapping them onto a specific lattice structure using these algorithms is equivalent to maximizing the likelihood that the network is generated by the models proposed. -We show that by calibrating a given network to both models, it is possible to quantify the relative presence of periodic and linear hierarchical structures using a likelihood ratio.…”