[Paper Review] The Embedded Homology of Hypergraphs and Applications

Stéphane Bressan, J. Li+2 more|arXiv (Cornell University)|Oct 4, 2016

Topological and Geometric Data Analysis8 references4 citations

TL;DR

This paper introduces embedded homology for hypergraphs, generalizing simplicial complex homology, and develops persistent embedded homology for sequences of hypergraphs. It establishes a Mayer-Vietoris sequence for embedded homology and applies the framework to define topological indices—hyper-network differentiation and correlation indices—for data analysis in hyper-networks.

ABSTRACT

Hypergraphs are mathematical models for many problems in data sciences. In recent decades, the topological properties of hypergraphs have been studied and various kinds of (co)homologies have been constructed (cf. [3, 4, 12]). In this paper, generalising the usual homology of simplicial complexes, we define the embedded homology of hypergraphs as well as the persistent embedded homology of sequences of hypergraphs. As a generalisation of the Mayer-Vietoris sequence for the homology of simplicial complexes, we give a Mayer-Vietoris sequence for the embedded homology of hypergraphs. Moreover, as applications of the embedded homology, we study acyclic hypergraphs and construct some indices for the data analysis of hyper-networks.

Motivation & Objective

• To generalize simplicial complex homology to hypergraphs via an embedded homology construction.
• To define persistent embedded homology for sequences of hypergraphs, extending persistent homology to hypergraph settings.
• To establish a Mayer-Vietoris sequence for embedded homology, generalizing the classical version for simplicial complexes.
• To characterize acyclic hypergraphs using the associated simplicial complex and embedded homology.
• To construct topological indices—differentiation and correlation indices—for data analysis in hyper-networks.

Proposed method

• Define the embedded homology of a hypergraph using the associated simplicial complex from Parks and Lipscomb (1991), which embeds the hypergraph into a simplicial complex.
• Construct the persistent embedded homology of a sequence of hypergraphs by filtering hyperedges based on thresholds and tracking homology evolution.
• Generalize the Mayer-Vietoris sequence to embedded homology by decomposing hypergraphs into sub-hypergraphs and relating their homology groups.
• Use the $L^2$-norm and degree of fitness to compare homology barcodes, defining a fitness measure between functions on hypergraphs.
• Define the hyper-network differentiation index as a measure of social or attribute variation, based on homology dimension changes across thresholds.
• Define the hyper-network correlation index by comparing 2D barcodes of homology dimensions under joint thresholds of two functions, using expected fitness against randomized controls.

Experimental results

Research questions

• RQ1 How can the homology of simplicial complexes be generalized to hypergraphs in a way that preserves topological structure?
• RQ2 Can a Mayer-Vietoris sequence be constructed for embedded homology of hypergraphs, analogous to the classical version for simplicial complexes?
• RQ3 How can topological invariants from embedded homology be used to characterize acyclic hypergraphs?
• RQ4 What topological indices can be derived from embedded homology to analyze data in hyper-networks?
• RQ5 How can the correlation between two vertex functions on a hypergraph be quantified using persistent homology and barcode comparison?

Key findings

• The embedded homology of a hypergraph coincides with the standard homology when the hypergraph is a simplicial complex.
• A Mayer-Vietoris sequence is established for embedded homology, enabling decomposition-based homology computation in hypergraphs.
• The persistent embedded homology of a sequence of hypergraphs stabilizes as the number of thresholds increases, due to the finiteness of the vertex set.
• The hyper-network differentiation index $\text{Diff}(\varphi,\mathcal{H})$ quantifies social differentiation or mobility, with decreasing values indicating reduced differentiation.
• The hyper-network correlation index $\text{Corr}(\varphi,\psi,\mathcal{H})$ measures the correlation between two vertex functions, with increasing values indicating stronger correlation under hyperedge structure.
• The correlation index is a real number between 0 and 1, derived from a weighted sum of fitness scores between 2D barcode functions and their randomized expectations.