An iterative spectral algorithm for digraph clustering

TL;DR

This paper introduces an iterative spectral algorithm for clustering directed graphs that preserves directional information and outperforms existing methods on synthetic and real-world data, also revealing higher-order cluster relations.

Contribution

The paper proposes a novel spectral clustering algorithm for directed graphs using Hermitian representations, effectively capturing directionality and higher-order relations.

Findings

Performs favorably against state-of-the-art methods

Successfully identifies higher-order meta-graph structures

Effective on diverse real-world datasets

Abstract

Graph clustering is a fundamental technique in data analysis with applications in many different fields. While there is a large body of work on clustering undirected graphs, the problem of clustering directed graphs is much less understood. The analysis is more complex in the directed graph case for two reasons: the clustering must preserve directional information in the relationships between clusters, and directed graphs have non-Hermitian adjacency matrices whose properties are less conducive to traditional spectral methods. Here we consider the problem of partitioning the vertex set of a directed graph into $k\ge 2$ clusters so that edges between different clusters tend to follow the same direction. We present an iterative algorithm based on spectral methods applied to new Hermitian representations of directed graphs. Our algorithm performs favourably against the state-of-the-art, both on synthetic and real-world data sets. Additionally, it is able to identify a "meta-graph" of $k$ vertices that represents the higher-order relations between clusters in a directed graph. We showcase this capability on data sets pertaining food webs, biological neural networks, and the online card game Hearthstone.