Link Partitioning on Simplicial Complexes Using Higher-Order Laplacians

TL;DR

This paper introduces a novel link partitioning method for simplicial complexes that leverages higher-order Laplacians and random walks to improve community detection, outperforming existing graph-based methods.

Contribution

It presents a new higher-order Laplacian-based approach for link partitioning in simplicial complexes, enabling better community detection with theoretical guarantees.

Findings

Significantly outperforms existing graph-based link partitioning methods on real-world data.

Provides new spectral insights into the properties of simplicial complexes.

Guarantees interpretability of link partitions under mild assumptions.

Abstract

Link partitioning is a popular approach in network science used for discovering overlapping communities by identifying clusters of strongly connected links. Current link partitioning methods are specifically designed for networks modelled by graphs representing pairwise relationships. Therefore, these methods are incapable of utilizing higher-order information about group interactions in network data which is increasingly available. Simplicial complexes extend the dyadic model of graphs and can model polyadic relationships which are ubiquitous and crucial in many complex social and technological systems. In this paper, we introduce a link partitioning method that leverages higher-order (i.e. triadic and higher) information in simplicial complexes for better community detection. Our method utilizes a novel random walk on links of simplicial complexes defined by the higher-order Laplacian--a generalization of the graph Laplacian that incorporates polyadic relationships of the network. We transform this random walk into a graph-based random walk on a lifted line graph--a dual graph in which links are nodes while nodes and higher-order connections are links--and optimize for the standard notion of modularity. We show that our method is guaranteed to provide interpretable link partitioning results under mild assumptions. We also offer new theoretical results on the spectral properties of simplicial complexes by studying the spectrum of the link random walk. Experiment results on real-world community detection tasks show that our higher-order approach significantly outperforms existing graph-based link partitioning methods.