Directed Network Laplacians and Random Graph Models

TL;DR

This paper explores spectral methods for detecting hidden structures in directed networks by connecting node reordering algorithms with directed random graph models, enabling comparison and analysis of network hierarchies.

Contribution

It establishes a link between spectral node reordering algorithms and directed random graph models, allowing for structure detection and comparison in directed networks.

Findings

Reordering algorithms relate to new classes of directed random graph models.

The approach enables quantification of the likelihood of different network structures.

Illustrations on synthetic and real networks demonstrate practical utility.

Abstract

We consider spectral methods that uncover hidden structures in directed networks. We establish and exploit connections between node reordering via (a) minimizing an objective function and (b) maximizing the likelihood of a random graph model. We focus on two existing spectral approaches that build and analyse Laplacian-style matrices via the minimization of frustration and trophic incoherence. These algorithms aim to reveal directed periodic and linear hierarchies, respectively. We show that reordering nodes using the two algorithms, or mapping them onto a specified lattice, is associated with new classes of directed random graph models. Using this random graph setting, we are able to compare the two algorithms on a given network and quantify which structure is more likely to be present. We illustrate the approach on synthetic and real networks, and discuss practical implementation issues.