Clustering with Simplicial Complexes
This work addresses clustering in networks for researchers and practitioners by leveraging higher-order structures, but it appears incremental as it builds on existing spectral clustering methods.
The paper tackles the problem of clustering nodes in networks by using second-order simplices (filled triangles) to capture higher-order interactions, proposing a simplicial spectral clustering algorithm that extends the Cheeger inequality and demonstrates efficacy on synthetic and real-world data.
In this work, we propose a new clustering algorithm to group nodes in networks based on second-order simplices (aka filled triangles) to leverage higher-order network interactions. We define a simplicial conductance function, which on minimizing, yields an optimal partition with a higher density of filled triangles within the set while the density of filled triangles is smaller across the sets. To this end, we propose a simplicial adjacency operator that captures the relation between the nodes through second-order simplices. This allows us to extend the well-known Cheeger inequality to cluster a simplicial complex. Then, leveraging the Cheeger inequality, we propose the simplicial spectral clustering algorithm. We report results from numerical experiments on synthetic and real-world network data to demonstrate the efficacy of the proposed approach.