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This study introduces a novel filtration method using persistent homology for cluster analysis in weighted graphs. The new approach offers richer topological signatures and improved robustness to outliers, enhancing graph data exploration.

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Area of Science:

  • Computational Topology
  • Graph Theory
  • Data Analysis

Background:

  • Persistent homology is a key tool for analyzing topological features of weighted graphs, particularly their 0-dimensional homology.
  • Existing methods often provide limited signatures for connected components and can be sensitive to noise.
  • There is a need for advanced filtration techniques in persistent homology for robust cluster analysis.

Purpose of the Study:

  • To develop a new filtration method for cluster analysis using persistent homology on weighted graphs.
  • To introduce non-trivial birth times for richer topological signatures of connected components.
  • To enhance robustness against outliers in graph-based cluster analysis.

Main Methods:

  • A novel filtration is constructed for weighted graphs based on persistent homology.
  • The method prioritizes nodes that become part of sufficiently large clusters, effectively ignoring outliers initially.
  • The approach focuses on 0-dimensional homology to capture connectivity patterns.

Main Results:

  • The new filtration yields richer topological signatures for connected components by incorporating non-trivial birth times.
  • The proposed method demonstrates significant robustness to outliers in the data.
  • Computational efficiency and practical effectiveness were demonstrated on random graphs.

Conclusions:

  • The developed persistent homology filtration offers an advanced tool for cluster analysis in weighted graphs.
  • Its ability to provide richer signatures and outlier robustness makes it valuable for complex network analysis.
  • The method shows promise for applications in various fields dealing with graph-structured data.