Persistent homology of hypergraphs via barycentric subdivision

Abstract Persistent homology is a mathematical tool used for studying the shape of data by extracting its topological features. It has gained popularity in network science due to its applicability in various network mining problems, including clustering, graph classification, and graph neural networks. Defining persistent homology for graphs is relatively straightforward, as graphs possess distinct intrinsic distances and a simplicial complex structure. However, hypergraphs present a challenge in preserving topological information since they may not have a simplicial complex structure. In this paper, we define two persistent homology filtrations for hypergraphs using their barycentric subdivision in defining persistent homology to extract different topological features within hypergraphs. To showcase the effectiveness, we employ these features in the hypergraph classification problem on four different real-world hypergraphs. We also compare their performance to the widely used simplicial complex closure filtration and also graph neural network models. Experimental results demonstrate that our persistent homology filtrations extract meaningful topological features that are effective in classifying hypergraphs and outperform the baseline models.