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Generalizing Weisfeiler-Lehman Kernels to Subgraphs

Dongkwan Kim, Alice Oh

TL;DR

This work tackles subgraph-level representation learning by addressing the expressiveness gaps of traditional GNNs. It introduces WLKS, a Weisfeiler-Lehman kernel generalized to subgraphs via induced $k$-hop neighborhoods, and merges information across $k$-hops, notably using $k\in\{0,D\}$ to balance expressiveness and efficiency. The authors prove that multi-$k$ combinations can be more expressive than any single $k$, and demonstrate that the kernel is PSD. Empirically, WLKS outperforms several state-of-the-art baselines on five of eight datasets while offering substantial training-time advantages and avoiding GPUs or pretraining, with extensions to continuous features and potential GNN integrations. This provides a practical, scalable alternative to deep GNNs for subgraph-level tasks, with code available for community use.

Abstract

Subgraph representation learning has been effective in solving various real-world problems. However, current graph neural networks (GNNs) produce suboptimal results for subgraph-level tasks due to their inability to capture complex interactions within and between subgraphs. To provide a more expressive and efficient alternative, we propose WLKS, a Weisfeiler-Lehman (WL) kernel generalized for subgraphs by applying the WL algorithm on induced $k$-hop neighborhoods. We combine kernels across different $k$-hop levels to capture richer structural information that is not fully encoded in existing models. Our approach can balance expressiveness and efficiency by eliminating the need for neighborhood sampling. In experiments on eight real-world and synthetic benchmarks, WLKS significantly outperforms leading approaches on five datasets while reducing training time, ranging from 0.01x to 0.25x compared to the state-of-the-art.

Generalizing Weisfeiler-Lehman Kernels to Subgraphs

TL;DR

This work tackles subgraph-level representation learning by addressing the expressiveness gaps of traditional GNNs. It introduces WLKS, a Weisfeiler-Lehman kernel generalized to subgraphs via induced -hop neighborhoods, and merges information across -hops, notably using to balance expressiveness and efficiency. The authors prove that multi- combinations can be more expressive than any single , and demonstrate that the kernel is PSD. Empirically, WLKS outperforms several state-of-the-art baselines on five of eight datasets while offering substantial training-time advantages and avoiding GPUs or pretraining, with extensions to continuous features and potential GNN integrations. This provides a practical, scalable alternative to deep GNNs for subgraph-level tasks, with code available for community use.

Abstract

Subgraph representation learning has been effective in solving various real-world problems. However, current graph neural networks (GNNs) produce suboptimal results for subgraph-level tasks due to their inability to capture complex interactions within and between subgraphs. To provide a more expressive and efficient alternative, we propose WLKS, a Weisfeiler-Lehman (WL) kernel generalized for subgraphs by applying the WL algorithm on induced -hop neighborhoods. We combine kernels across different -hop levels to capture richer structural information that is not fully encoded in existing models. Our approach can balance expressiveness and efficiency by eliminating the need for neighborhood sampling. In experiments on eight real-world and synthetic benchmarks, WLKS significantly outperforms leading approaches on five datasets while reducing training time, ranging from 0.01x to 0.25x compared to the state-of-the-art.

Paper Structure

This paper contains 34 sections, 2 theorems, 4 equations, 5 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Subgraph Representation Learning
  4. Graph Kernels
  5. WL Graph Kernels for Subgraph-Level Tasks
  6. Subgraph Representation Learning
  7. 1-WL Algorithm for $k$-hop Subgraphs
  8. 1-WL for Graphs
  9. 1-WL for Subgraphs (WLS)
  10. WL Kernels for Subgraphs (WLKS)
  11. Expressiveness difference of the WLS between $k$ and $k+1$
  12. Proof of Equation \ref{['eq:wls_main_1']}
  13. Proof of Equation \ref{['eq:wls_main_2']}
  14. Selecting $k$ for Minimal Complexity
  15. Computational Complexity
  16. ...and 19 more sections

Key Result

Proposition 3.1

$\forall k \geq 0, {\bm{K}}_{\mathtt{WLS}}^{k}$ is positive semi-definite (p.s.d.).

Figures (5)

  • Figure 1: An example of $\mathtt{WLS}^k$ algorithm (Algorithm \ref{['alg:wlks']}) for $k \in \{ 0, 1, 2 \}$. Left: A subgraph (red shade) and its $k$-hop neighborhoods (dashed lines). Right: The outputs of $\mathtt{WLS}^k$ algorithm as colors and histograms for the left subgraph. We visualize each iteration of the algorithm in Appendix \ref{['appendix:model_steps']}. The WLKS kernel matrix for each $k$ is constructed by an inner product of histogram pairs.
  • Figure 2: Example pairs of subgraphs that $\mathtt{WLS}^{k}$ produces equivalent colorings while $\mathtt{WLS}^{k+1}$ does not, and vice versa (where $k=0$). The gray area represents the subgraph.
  • Figure 3: Performance of WLKS-$\{0,D\}$ by the number of iterations $T$.
  • Figure 4: Performance of WLKS-$\{0,D\}$ by the coefficient $\alpha_0$ in $\alpha_{0} {\bm{K}}_{\mathtt{WLS}}^{0} + (1 - \alpha_{0}) {\bm{K}}_{\mathtt{WLS}}^{D}$.
  • Figure 5: A step-by-step visualization of $\mathtt{WLS}^k$ algorithm (Algorithm \ref{['alg:wlks']}) for $k \in \{ 0, 1, 2 \}$ using an example in Figure \ref{['fig:model']}.

Theorems & Definitions (4)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof