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Towards Subgraph Isomorphism Counting with Graph Kernels

Xin Liu, Weiqi Wang, Jiaxin Bai, Yangqiu Song

TL;DR

This work addresses the challenge of subgraph isomorphism counting, a #P-complete problem, by reframing counting as a regression task over graph kernels. It introduces neighborhood-information-extraction (NIE) to enrich WL-based kernel representations and constructs a Gram-matrix-based framework that leverages implicit correlations to predict subgraph counts. The authors explore kernel tricks, including polynomial and Gaussian kernels, and develop a hybrid NIE-WL kernel that decomposes into WL plus NIE components, enabling efficient implicit feature mappings. Empirical results across synthetic and real datasets show that NIE generally improves performance, with Gaussian kernels performing strongly on homogeneous data and linear kernels excelling on heterogeneous data; the approach compares favorably with some neural methods on challenging benchmarks. The findings highlight the potential of graph kernels, especially with NIE and kernel tricks, as scalable, interpretable tools for approximate subgraph counting and motivate integrating such representations with advanced graph learning techniques in future work.

Abstract

Subgraph isomorphism counting is known as #P-complete and requires exponential time to find the accurate solution. Utilizing representation learning has been shown as a promising direction to represent substructures and approximate the solution. Graph kernels that implicitly capture the correlations among substructures in diverse graphs have exhibited great discriminative power in graph classification, so we pioneeringly investigate their potential in counting subgraph isomorphisms and further explore the augmentation of kernel capability through various variants, including polynomial and Gaussian kernels. Through comprehensive analysis, we enhance the graph kernels by incorporating neighborhood information. Finally, we present the results of extensive experiments to demonstrate the effectiveness of the enhanced graph kernels and discuss promising directions for future research.

Towards Subgraph Isomorphism Counting with Graph Kernels

TL;DR

This work addresses the challenge of subgraph isomorphism counting, a #P-complete problem, by reframing counting as a regression task over graph kernels. It introduces neighborhood-information-extraction (NIE) to enrich WL-based kernel representations and constructs a Gram-matrix-based framework that leverages implicit correlations to predict subgraph counts. The authors explore kernel tricks, including polynomial and Gaussian kernels, and develop a hybrid NIE-WL kernel that decomposes into WL plus NIE components, enabling efficient implicit feature mappings. Empirical results across synthetic and real datasets show that NIE generally improves performance, with Gaussian kernels performing strongly on homogeneous data and linear kernels excelling on heterogeneous data; the approach compares favorably with some neural methods on challenging benchmarks. The findings highlight the potential of graph kernels, especially with NIE and kernel tricks, as scalable, interpretable tools for approximate subgraph counting and motivate integrating such representations with advanced graph learning techniques in future work.

Abstract

Subgraph isomorphism counting is known as #P-complete and requires exponential time to find the accurate solution. Utilizing representation learning has been shown as a promising direction to represent substructures and approximate the solution. Graph kernels that implicitly capture the correlations among substructures in diverse graphs have exhibited great discriminative power in graph classification, so we pioneeringly investigate their potential in counting subgraph isomorphisms and further explore the augmentation of kernel capability through various variants, including polynomial and Gaussian kernels. Through comprehensive analysis, we enhance the graph kernels by incorporating neighborhood information. Finally, we present the results of extensive experiments to demonstrate the effectiveness of the enhanced graph kernels and discuss promising directions for future research.
Paper Structure (29 sections, 10 equations, 6 figures, 5 tables, 2 algorithms)

This paper contains 29 sections, 10 equations, 6 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Background and Motivations
  4. Problem Definition
  5. Graph Isomorphism Test and Representation Power
  6. Graph Kernels
  7. Beyond the Representation Power Limitation via Implicit Correlations
  8. Gram Matrix Construction
  9. Neighborhood Information in the Hilbert Space
  10. Kernel Tricks
  11. Polynomial Kernel
  12. Gaussian Kernel
  13. Experiment
  14. Experimental Setup
  15. Evaluation
  16. ...and 14 more sections

Figures (6)

  • Figure 1: Examples of the isomorphism and subgraph isomorphism, where letters indicate labels.
  • Figure 2: Example of the non-isomorphic graphs with the same WL color histogram but different 3-WL color histograms.
  • Figure 3: Four patterns considered in experiments: (a) 3-star, (b) triangle, (c) tailed triangle, and (d) chordal cycle.
  • Figure 4: Efficient implementation.
  • Figure 5: Illustration on subgraph isomorphism counting, where the out-of-memory ("OOM") is regarded as zero in the plots.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Definition 3.1: Isomorphism
  • Definition 3.2: Subgraph isomorphism
  • Definition 3.3: Graph Isomorphism Test
  • Definition 3.4: Kernel