Reduction Algorithms for Persistence Diagrams of Networks: CoralTDA and PrunIT

TL;DR

This paper introduces two simple yet effective algorithms, CoralTDA and PrunIT, that significantly reduce the computational costs of calculating persistence diagrams for large networks, enabling TDA applications in big data scenarios.

Contribution

The paper presents novel algorithms that leverage graph core decomposition and vertex pruning to compute exact persistence diagrams efficiently for large-scale networks.

Findings

Achieved up to 95% reduction in computation time.

Proved that the (k+1)-core suffices for computing the k-th persistence diagram.

First framework connecting graph theory with TDA for large networks.

Abstract

Topological data analysis (TDA) delivers invaluable and complementary information on the intrinsic properties of data inaccessible to conventional methods. However, high computational costs remain the primary roadblock hindering the successful application of TDA in real-world studies, particularly with machine learning on large complex networks. Indeed, most modern networks such as citation, blockchain, and online social networks often have hundreds of thousands of vertices, making the application of existing TDA methods infeasible. We develop two new, remarkably simple but effective algorithms to compute the exact persistence diagrams of large graphs to address this major TDA limitation. First, we prove that $(k+1)$-core of a graph $\mathcal{G}$ suffices to compute its $k^{th}$ persistence diagram, $PD_k(\mathcal{G})$. Second, we introduce a pruning algorithm for graphs to compute their persistence diagrams by removing the dominated vertices. Our experiments on large networks show that our novel approach can achieve computational gains up to 95%. The developed framework provides the first bridge between the graph theory and TDA, with applications in machine learning of large complex networks. Our implementation is available at https://github.com/cakcora/PersistentHomologyWithCoralPrunit