Spectral clustering algorithms for the detection of clusters in block-cyclic and block-acyclic graphs

TL;DR

This paper introduces two spectral clustering algorithms tailored for directed graphs with cyclic and acyclic structures, outperforming existing methods in synthetic datasets and revealing meaningful patterns in real-world networks.

Contribution

The paper presents novel spectral algorithms for directed graph clustering based on extremal eigenvalues, with linear complexity and successful applications to ecological and internet networks.

Findings

Outperform state-of-the-art methods on synthetic datasets

Successfully identify hierarchical and cyclic structures in real networks

Maintain linear time complexity similar to classical spectral clustering

Abstract

We propose two spectral algorithms for partitioning nodes in directed graphs respectively with a cyclic and an acyclic pattern of connection between groups of nodes. Our methods are based on the computation of extremal eigenvalues of the transition matrix associated to the directed graph. The two algorithms outperform state-of-the art methods for directed graph clustering on synthetic datasets, including methods based on blockmodels, bibliometric symmetrization and random walks. Our algorithms have the same space complexity as classical spectral clustering algorithms for undirected graphs and their time complexity is also linear in the number of edges in the graph. One of our methods is applied to a trophic network based on predator-prey relationships. It successfully extracts common categories of preys and predators encountered in food chains. The same method is also applied to highlight the hierarchical structure of a worldwide network of Autonomous Systems depicting business agreements between Internet Service Providers.