Persistent homology filtration grounded on higher-order complex networks centrality measures

Area of Science:

• Network Science
• Computational Topology
• Data Analysis

Background:

• Analyzing complex networks often involves pairwise interactions, overlooking higher-order structures.
• Integrating higher-order network analysis with traditional methods presents significant challenges.
• Persistent homology offers a computational approach to understand topological features in simplicial complexes.

Purpose of the Study:

• To develop a consistent methodological framework for analyzing higher-order structures in complex networks.
• To generalize network measures and apply them within a persistent homology framework.
• To demonstrate the utility of the proposed methods on real-world network data.

Main Methods:

• Derivation of a weighted simplicial adjacency matrix applicable to generalized network measures.
• Development of three distinct filtration schemes for constructing simplicial complexes based on generalized measures.
• Application of persistent homology to compute topological features (Betti numbers) of network structures.

Main Results:

• The proposed framework successfully integrates pairwise and higher-order network analysis.
• Generalized measures serve as effective filtration parameters for persistent homology.
• The method reveals hidden layers and topological patterns in complex networks, illustrated with Betti number computation.

Conclusions:

• The developed approach provides a robust method for analyzing higher-order network structures.
• This unified framework enhances the comprehensive understanding of complex systems by incorporating topological features.
• The application to real-world networks validates the effectiveness and benefits of the established methodology.