Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explaining the Power of Topological Data Analysis in Graph Machine Learning

Funmilola Mary Taiwo, Umar Islambekov, Cuneyt Gurcan Akcora

TL;DR

This work rigorously evaluates the claims of Topological Data Analysis for graph machine learning by benchmarking persistence-based methods against Weisfeiler-Leman kernels and traditional graph features across nine graph-classification datasets. It finds that while TDA delivers robustness and interpretable insights, it does not consistently improve predictive accuracy and incurs substantial computational costs, especially for Vietoris-Rips filtrations. Betti-function vectorizations emerge as a relatively strong TDA signal, with most informative topology occurring at early filtration ranges, and a surrogate model plus Mapper analyses provide explainable avenues to harness TDA signals. The paper suggests using TDA as a complementary, interpretable layer within graph pipelines and emphasizes strategies to identify informative filtration ranges to enhance scalability and practical deployment.

Abstract

Topological Data Analysis (TDA) has been praised by researchers for its ability to capture intricate shapes and structures within data. TDA is considered robust in handling noisy and high-dimensional datasets, and its interpretability is believed to promote an intuitive understanding of model behavior. However, claims regarding the power and usefulness of TDA have only been partially tested in application domains where TDA-based models are compared to other graph machine learning approaches, such as graph neural networks. We meticulously test claims on TDA through a comprehensive set of experiments and validate their merits. Our results affirm TDA's robustness against outliers and its interpretability, aligning with proponents' arguments. However, we find that TDA does not significantly enhance the predictive power of existing methods in our specific experiments, while incurring significant computational costs. We investigate phenomena related to graph characteristics, such as small diameters and high clustering coefficients, to mitigate the computational expenses of TDA computations. Our results offer valuable perspectives on integrating TDA into graph machine learning tasks.

Explaining the Power of Topological Data Analysis in Graph Machine Learning

TL;DR

This work rigorously evaluates the claims of Topological Data Analysis for graph machine learning by benchmarking persistence-based methods against Weisfeiler-Leman kernels and traditional graph features across nine graph-classification datasets. It finds that while TDA delivers robustness and interpretable insights, it does not consistently improve predictive accuracy and incurs substantial computational costs, especially for Vietoris-Rips filtrations. Betti-function vectorizations emerge as a relatively strong TDA signal, with most informative topology occurring at early filtration ranges, and a surrogate model plus Mapper analyses provide explainable avenues to harness TDA signals. The paper suggests using TDA as a complementary, interpretable layer within graph pipelines and emphasizes strategies to identify informative filtration ranges to enhance scalability and practical deployment.

Abstract

Topological Data Analysis (TDA) has been praised by researchers for its ability to capture intricate shapes and structures within data. TDA is considered robust in handling noisy and high-dimensional datasets, and its interpretability is believed to promote an intuitive understanding of model behavior. However, claims regarding the power and usefulness of TDA have only been partially tested in application domains where TDA-based models are compared to other graph machine learning approaches, such as graph neural networks. We meticulously test claims on TDA through a comprehensive set of experiments and validate their merits. Our results affirm TDA's robustness against outliers and its interpretability, aligning with proponents' arguments. However, we find that TDA does not significantly enhance the predictive power of existing methods in our specific experiments, while incurring significant computational costs. We investigate phenomena related to graph characteristics, such as small diameters and high clustering coefficients, to mitigate the computational expenses of TDA computations. Our results offer valuable perspectives on integrating TDA into graph machine learning tasks.
Paper Structure (26 sections, 7 equations, 12 figures, 5 tables)

This paper contains 26 sections, 7 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Research Questions
  4. Persistence-based Topological Methods
  5. Vietoris-Rips Complexes
  6. Alpha Complexes
  7. Filtration
  8. Vectorizations of Persistence Diagrams
  9. Experimental Setup
  10. Answers to Research Questions
  11. Power and Components of TDA
  12. Complementary Power of TDA
  13. Scalability of TDA
  14. Robustness of TDA
  15. Interpretability of TDA
  16. ...and 11 more sections

Figures (12)

  • Figure 1: A Topology-Based View on Datasets. The Center Contains $\texttt{NCI1}$ Graphs.
  • Figure 2: Vietoris-Rips Filtration. A Weighted Graph $\mathcal{G}$ is Embedded Into a Metric Space where the Distance Between Two Nodes is Defined as the Shortest Path Between Them. $\mathcal{G}_1,\ldots,\mathcal{G}_4$ are the First Four Complexes of the Resulting Vietoris-Rips Filtration, where the Nodes are 0-Simplices, the Edges are 1-Simplices and (Filled) Triangles are 2-simplices. At $\epsilon=3$, a One-Dimensional Hole is Formed by Edges $C-E$, $C-D$, $D-F$, and $E-F$, but gets Filled at $\epsilon=4$ when Triangles (2-Simplices) $\triangle{CDE}$ and $\triangle{DEF}$ are added to $G_4$.
  • Figure 3: Variation in Accuracy as a Function of the Percentage of Deleted Edges in a Graph, Averaged Over the Nine Datasets. TDA-Based Vietoris-Rips (VR) Model in Panel C Exhibits Better Decay Rates. For All Three Models, ENZYMES is the Outlier Dataset.
  • Figure 4: Useful Filtration Ranges. The x-axis Values are the Filtration Thresholds (in $[0,100]$) Defined Over Shortest Path Distance Between Node Pairs, and a Longer Range Between the Minimum and Maximum Values Indicates Greater Variability and Importance of these Thresholds in Influencing the Model's Predictions. The Range in figure \ref{['fig:b1']} is Between the Minimum and Maximum Values of the 10 Most Important Filtration Thresholds identified through the Shapley analysis.
  • Figure 5: Decision Tree Showing Features Split Using Decision Tree Regressor When the Target is Betti 1.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Definition 1: Abstract Simplicial Complex
  • Definition 2: Vietoris-Rips Complex vietoris1927hoheren
  • Definition 3: Alpha Complex
  • Definition 4: Filtration
  • Definition 5: Betti Function
  • Definition 6: Persistence Landscape (PL)
  • Definition 7: Persistence Silhouette (PS)
  • Definition 8: Weisfeiler-Leman Kernel