[Paper Review] Topological Brain Network Distances

Moo K. Chung, Hyekyoung Lee+5 more|arXiv (Cornell University)|Sep 9, 2018

Topological and Geometric Data Analysis12 references3 citations

TL;DR

This paper introduces topological brain network distances—bottleneck, Gromov-Hausdorff (GH), and Kolmogorov-Smirnov (KS)—to overcome limitations of matrix-norm-based distances that fail to capture topological features like connected components and cycles. The KS distance outperforms others in detecting heritability in twin fMRI data, identifying 100% heritable connections in regions including the left and right caudate nuclei and frontal gyri.

ABSTRACT

Existing brain network distances are often based on matrix norms. The element-wise differences in the existing matrix norms may fail to capture underlying topological differences. Further, matrix norms are sensitive to outliers. A major disadvantage to element-wise distance calculations is that it could be severely affected even by a small number of extreme edge weights. Thus it is necessary to develop network distances that recognize topology. In this paper, we provide a survey of bottleneck, Gromov-Hausdorff (GH) and Kolmogorov-Smirnov (KS) distances that are adapted for brain networks, and compare them against matrix-norm based network distances. Bottleneck and GH-distances are often used in persistent homology. However, they were rarely utilized to measure similarity between brain networks. KS-distance is recently introduced to measure the similarity between networks across different filtration values. The performance analysis was conducted using the random network simulations with the ground truths. Using a twin imaging study, which provides biological ground truth, we demonstrate that the KS distance has the ability to determine heritability.

Motivation & Objective

• To address the insensitivity of matrix-norm-based network distances to topological features such as connected components, modules, and holes in brain networks.
• To overcome the limitations of arbitrary thresholding in binarizing weighted brain networks by avoiding fixed thresholds.
• To develop robust, topology-aware network distances that are invariant to noise and sensitive to persistent topological structures across scales.
• To evaluate the performance of topological distances (KS, GH, bottleneck) against matrix-norm-based distances using simulated and real twin fMRI data.
• To demonstrate the utility of the KS distance in detecting biological heritability in brain connectivity networks.

Proposed method

• Adapts bottleneck and Gromov-Hausdorff (GH) distances from persistent homology to measure topological similarity between brain networks.
• Introduces the Kolmogorov-Smirnov (KS) distance as a novel method to compare cumulative distribution functions of topological features (e.g., Betti numbers) across different filtration values.
• Uses graph filtration to generate a nested family of subgraphs across all edge weight thresholds, enabling multi-scale topological analysis without arbitrary thresholding.
• Employs persistent homology to track topological features (e.g., connected components β₀, loops β₁) across filtration levels, forming the basis for distance computation.
• Applies the KS distance to compare the empirical distribution functions of Betti numbers across networks, avoiding computationally expensive permutation tests.
• Uses permutation-based inference for matrix-norm and GH distances, while deriving combinatorial p-values for KS distance to avoid resampling.

Experimental results

Research questions

• RQ1 Can topological distances such as KS, GH, and bottleneck better capture structural differences in brain networks than traditional matrix-norm-based distances?
• RQ2 Does the KS distance outperform other topological and matrix-based distances in detecting heritability in twin fMRI data?
• RQ3 Can topological distances detect persistent network features across multiple scales without relying on arbitrary thresholding?
• RQ4 How do topological distances perform in distinguishing networks with identical single-linkage matrices but different cycle structures?
• RQ5 Can the KS distance avoid the computational burden of permutation testing while maintaining statistical validity?

Key findings

• The KS distance successfully identified 100% heritability in brain network connections, with the highest heritability observed in the left and right caudate nuclei, left and right middle frontal gyri, and left superior frontal gyrus.
• The most heritable connections, including those involving the left and right thalami and parahippocampal gyri, align with previously reported heritable regions in resting-state fMRI and DTI studies.
• The KS distance demonstrated superior performance in detecting biologically meaningful network differences compared to matrix-norm-based distances, particularly in the presence of noise and outliers.
• Unlike GH-distance, which fails to distinguish networks with identical single-linkage matrices but different cycle structures, the KS distance can detect such differences when using β₁ (loop) features.
• The KS distance avoids computationally intensive permutation testing by computing p-values combinatorially, offering a scalable alternative to resampling-based inference.
• The study confirms that topological distances, especially KS, are more robust and informative than matrix-norm-based distances for analyzing brain network similarity and heritability.