Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Graph Persistence goes Spectral

Mattie Ji, Amauri H. Souza, Vikas Garg

TL;DR

This work introduces SpectRe, a spectrum-informed persistence descriptor for graphs that augments color-based persistent homology (via RePHINE) with Laplacian spectral information. SpectRe is proven to be strictly more expressive than both RePHINE and the Laplacian spectrum, while exhibiting local stability under a curated bottleneck-like metric. The authors detail integration of SpectRe with graph neural networks using per-layer vectorization through DeepSets, and validate the approach with extensive synthetic and real-world experiments, showing robust performance gains. The method broadens the toolkit for graph representation learning by fusing topological signals with spectral information, at the cost of increased computational complexity, which can be mitigated with approximations and scheduling.

Abstract

Including intricate topological information (e.g., cycles) provably enhances the expressivity of message-passing graph neural networks (GNNs) beyond the Weisfeiler-Leman (WL) hierarchy. Consequently, Persistent Homology (PH) methods are increasingly employed for graph representation learning. In this context, recent works have proposed decorating classical PH diagrams with vertex and edge features for improved expressivity. However, these methods still fail to capture basic graph structural information. In this paper, we propose SpectRe -- a new topological descriptor for graphs that integrates spectral information into PH diagrams. Notably, SpectRe is strictly more expressive than existing descriptors on graphs. We also introduce notions of global and local stability to analyze existing descriptors and establish that SpectRe is locally stable. Finally, experiments on synthetic and real-world datasets demonstrate the effectiveness of SpectRe and its potential to enhance the capabilities of graph models in relevant learning tasks. Code is available at https://github.com/Aalto-QuML/SpectRe/.

Graph Persistence goes Spectral

TL;DR

This work introduces SpectRe, a spectrum-informed persistence descriptor for graphs that augments color-based persistent homology (via RePHINE) with Laplacian spectral information. SpectRe is proven to be strictly more expressive than both RePHINE and the Laplacian spectrum, while exhibiting local stability under a curated bottleneck-like metric. The authors detail integration of SpectRe with graph neural networks using per-layer vectorization through DeepSets, and validate the approach with extensive synthetic and real-world experiments, showing robust performance gains. The method broadens the toolkit for graph representation learning by fusing topological signals with spectral information, at the cost of increased computational complexity, which can be mitigated with approximations and scheduling.

Abstract

Including intricate topological information (e.g., cycles) provably enhances the expressivity of message-passing graph neural networks (GNNs) beyond the Weisfeiler-Leman (WL) hierarchy. Consequently, Persistent Homology (PH) methods are increasingly employed for graph representation learning. In this context, recent works have proposed decorating classical PH diagrams with vertex and edge features for improved expressivity. However, these methods still fail to capture basic graph structural information. In this paper, we propose SpectRe -- a new topological descriptor for graphs that integrates spectral information into PH diagrams. Notably, SpectRe is strictly more expressive than existing descriptors on graphs. We also introduce notions of global and local stability to analyze existing descriptors and establish that SpectRe is locally stable. Finally, experiments on synthetic and real-world datasets demonstrate the effectiveness of SpectRe and its potential to enhance the capabilities of graph models in relevant learning tasks. Code is available at https://github.com/Aalto-QuML/SpectRe/.

Paper Structure

This paper contains 26 sections, 15 theorems, 50 equations, 4 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Persistent Homology on Graphs
  4. RePHINE
  5. Spectrum-informed Persistence Diagrams
  6. Laplacian Spectrum and SpectRe
  7. Expressive Power
  8. Stability of RePHINE and SpectRe
  9. Integration with GNNs
  10. Experiments
  11. Synthetic data
  12. Real data
  13. Conclusion
  14. Proofs
  15. Proofs for Section \ref{['sec::filtration']}
  16. ...and 11 more sections

Key Result

Theorem 3.2

Suppose $G$ and $H$ are isomorphic graphs with the same color set $X$. Let $f = (f_v, f_e)$ be any filtration functions on $X$, then $\operatorname{SpectRe}(G, f)$ is equal to $\operatorname{SpectRe}(H, f)$.

Figures (4)

  • Figure 1: Vertex-level PH: filtration and diagram induced by $f_v$. Here, we have that $f_v(\operatorname{blue})=1$ and $f_v(\operatorname{red})=2$.
  • Figure 2: Example computing SpectRe, RePHINE, (edge-level) PH, and LS on a graph $G$ with $f_v(\operatorname{blue}) = 1, f_v(\operatorname{red}) = 2$ and $f_e(\operatorname{red}) = 1, f_e(\operatorname{blue}) = 2, f_e(\operatorname{red-blue}) = 3$. The graph $G$ has an edge filtration by $f_e$.
  • Figure 3: (a) Graphs that $\operatorname{SpectRe}$ and $\operatorname{LS}$ can separate but not $\operatorname{RePHINE}$. (b) Graphs that $\operatorname{SpectRe}$ and $\operatorname{RePHINE}$ can separate but not $\operatorname{LS}$.
  • Figure 4: Example of a pair of graphs from the dataset Extension(100) that RePHINE cannot distinguish but SpectRe can. Node and edge labels denote filtration values.

Theorems & Definitions (46)

  • Definition 2.1: Coloring Filtrations
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 3.4
  • Definition 4.2
  • Definition 4.3
  • Theorem 4.4
  • ...and 36 more