[Paper Review] Topological Distances between Networks and Its Application to Brain Imaging

Hyekyoung Lee, Zhiwei Ma+2 more|arXiv (Cornell University)|Jan 16, 2017

Topological and Geometric Data Analysis51 references4 citations

TL;DR

This paper introduces topological network distance measures—bottleneck, kernel, Gromov-Hausdorff, and a KS-test-like distance—based on persistent homology to quantify brain network differences without thresholding. It demonstrates that these methods detect subtle topological changes in brain networks across clinical populations more robustly than traditional graph metrics.

ABSTRACT

This paper surveys various distance measures for networks and graphs that were introduced in persistent homology. The scope of the paper is limited to network distances that were actually used in brain networks but the methods can be easily adapted to any weighted graph in other fields. The network version of Gromov-Hausdorff, bottleneck, kernel distances are introduced. We also introduce a recently developed KS-test like distance based on monotonic topology features such as the zeroth Betti number. Numerous toy examples and the result of applying many different distances to the brain networks of different clinical status and populations are given.

Motivation & Objective

• To develop threshold-free, topologically robust network distance measures for brain networks using persistent homology.
• To overcome limitations of conventional graph-theoretic features and threshold-dependent binarization in brain network analysis.
• To enable statistical inference on network topology using nonparametric, monotonic topological features such as the zeroth Betti number.
• To compare the performance of multiple topological distances (bottleneck, kernel, Gromov-Hausdorff, KS-like) on real brain network data.
• To provide a scalable, computationally feasible alternative to permutation-based inference for topological network distances.

Proposed method

• Uses graph filtration to generate a nested family of weighted subgraphs across all possible thresholds, avoiding arbitrary binarization.
• Applies persistent homology to track topological features (e.g., connected components, cycles) across scales, represented as persistence diagrams.
• Employs bottleneck distance to compare persistence diagrams by minimizing the maximum distance between matched points.
• Utilizes kernel distances via heat diffusion on persistence diagrams to enable Hilbert space operations and statistical inference.
• Adapts Gromov-Hausdorff distance to brain networks using node correspondence from image registration, reducing it to single linkage matrix comparison.
• Introduces a KS-test-like distance based on monotonic topological features (e.g., zeroth Betti number) for nonparametric, distribution-free inference.

Experimental results

Research questions

• RQ1 Can topological distances based on persistent homology detect meaningful differences in brain networks across clinical populations without thresholding?
• RQ2 How do different topological distances (bottleneck, kernel, Gromov-Hausdorff, KS-like) compare in sensitivity and robustness to noise and network structure?
• RQ3 Can monotonic topological features such as the zeroth Betti number support nonparametric statistical inference on network differences?
• RQ4 To what extent do traditional graph metrics fail to capture topological features like cycles or connected components compared to persistent homology?
• RQ5 Is the proposed KS-test-like distance computationally more efficient than permutation-based inference while maintaining statistical validity?

Key findings

• The KS-test-like distance based on monotonic topological features enables exact nonparametric inference without assuming underlying distributions.
• The method requires only up to 100 iterations for p-value estimation, significantly outperforming permutation-based methods that require tens of thousands of permutations.
• Persistent homology-based distances detect topological differences in brain networks more robustly than standard graph metrics, especially in detecting cycles and connected components.
• The Gromov-Hausdorff distance, when combined with node correspondence, effectively captures structural differences but fails to distinguish networks with identical single linkage matrices, such as those with cycles.
• Kernel distances based on heat diffusion on persistence diagrams enable downstream statistical analysis via Hilbert space tools like SVM and PCA.
• The proposed framework avoids the circularity of threshold selection by analyzing all possible thresholds simultaneously through graph filtration.