Graph Kernels

• Graph Kernels are positive semidefinite similarity functions that map graphs into high-dimensional Hilbert spaces via implicit or explicit feature maps.
• They employ methods such as R-convolution, random-walk, shortest-path, and Weisfeiler–Lehman techniques to compare graph substructures effectively.
• Widely used in chemoinformatics, bioinformatics, social network analysis, and computer vision, graph kernels balance theoretical rigor with practical scalability.

A graph kernel is a positive semidefinite function that measures the similarity between graphs by embedding them into a (possibly infinite-dimensional) Hilbert space and comparing inner products in that space (Kriege et al., 2019). The graph kernel paradigm enables kernel-based machine learning (e.g., SVM, kernel PCA) on structured data by circumventing the need for explicit vector representations. Over the past two decades, graph kernels have developed into a major methodology in graph-based learning, with numerous families tailored to distinct structural, geometric, and attribute-based properties.

1. Theoretical Foundations

Positive Semidefiniteness and Feature Maps

A graph kernel $$k: \mathcal{G} \times \mathcal{G} \rightarrow \mathbb{R}$$, with $\mathcal{G}$ a family of graphs (possibly with node/edge labels or attributes), is required to be symmetric and positive semidefinite:

$$k(G,H) = k(H,G) \qquad \sum_{i,j} c_i c_j\, k(G_i, G_j) \ge 0\;\; \forall\,c_i\in\mathbb{R},\,G_i\in\mathcal{G}$$

By Mercer’s theorem, there exists an implicit feature map $\phi: \mathcal{G} \to \mathcal{H}$ such that $$k(G,H) = \langle \phi(G), \phi(H) \rangle_{\mathcal{H}}$$. This “kernel trick” allows complex graph similarity to be assessed via inner products, often with explicit or implicit construction of the feature space (Nikolentzos et al., 2019).

R-Convolution Framework

Most graph kernels adhere to Haussler’s R-convolution scheme, decomposing graphs into substructures (walks, subtrees, cycles, graphlets, shortest paths, etc.), comparing substructures with a base kernel $k_{\mathrm{base}}$, and aggregating globally:

$$k(G,H) = \sum_{(g,h) \in R^{-1}(G) \times R^{-1}(H)} k_{\mathrm{base}}(g,h)$$

This includes walk kernels, subtree kernels, assignment kernels, and many others (Kriege et al., 2019).

2. Canonical Families of Graph Kernels

The major graph kernel paradigms can be categorized by the substructures they count or aggregate and the associated algorithms.

Kernel Family	Structural Motif	Complexity *
Random-Walk (RW)	Matching walks	$O(n^6)$ or $O(n^3)$
Shortest-Path (SP)	All-pairs shortest paths	$O(n^4)$
Graphlets	Induced k-node subgraphs	$O(n^k)$
Weisfeiler–Lehman (WL)	subtree/label refinement	$O(hm)$
Spectral/DOS/LDOS	Eigenvalue/global spectrum	$O(n^3)$ or $O(|E|)$

* $n = |V(G)|$, $m = |E(G)|$, $h$ = # WL iterations.

Random-Walk Kernels: Compare two graphs by counting matching label sequences along walks of all lengths via the adjacency matrix of their direct product $A_\times$. The geometric RW kernel is:

$$k_{\mathrm{RW}}(G,H) = \sum_{\ell=0}^\infty \gamma^\ell\, 1^\top A_\times^\ell 1 = 1^\top (I - \gamma A_\times)^{-1} 1$$

with $0 < \gamma < 1/\rho(A_\times)$. Efficient computation is achieved via Sylvester or Lyapunov reduction to $O(n^3)$ (0807.0093).

Shortest-Path Kernels: Rely on all-pairs distances $D_G(u,v)$. The kernel compares label and path-length triples:

$$k_{\mathrm{SP}}(G,H) = \sum_{u\neq v} \sum_{u'\neq v'} k_L(l(u),l(u'))\,k_L(l(v),l(v'))\,k_D(D_G(u,v),D_H(u',v'))$$

Efficient for discrete labels via explicit feature vector construction (Kriege et al., 2019).

Weisfeiler–Lehman (WL) Kernels: The h-iteration WL subtree kernel color-refines node labels by the multiset of neighbor colors, then counts occurrences. The kernel is

$$k_{\mathrm{WL}}(G,H) = \sum_{i=0}^h \langle \phi^i(G), \phi^i(H) \rangle,$$

where $\phi^i(G)$ is the histogram of labels at WL iteration $i$ (Nikolentzos et al., 2019).

Graphlet Kernels: Count $k$-node induced subgraphs of each isomorphism type, forming feature vectors $\phi(G)\in \mathbb{N}^d$, $d=$ number of distinct graphlets (Kriege et al., 2019).

Spectral (Density of States): Embeds a graph by the density of states $\mu(\lambda)$ or local DOS $\mu_k(\lambda)$ from the spectrum of the (normalized) adjacency matrix. DOS/LDOS kernels use moment features or Maximum Mean Discrepancy in the RKHS of empirical spectral distributions (Huang et al., 2020).

3. Expressivity, Efficiency, and Practical Considerations

Expressivity vs. Efficiency: There exists an inherent trade-off. Complete (isomorphism) kernels require intractable computation (e.g., counting all subgraphs), as shown by quantum kernels considering all $2^n$ induced subgraphs (Kishi et al., 2021). Standard kernels—WL, SP, graphlet—approximate local or mid-range structure with polynomial complexity. Higher-dimensional WL and assignment-based kernels lift discriminative power at cost of exponential runtime.

Scalability: WL and explicit-graphlet kernels scale to large graphs ($n>10^4$) due to $O(hm)$ time for WL and sampling strategies for graphlets (Kriege et al., 2019). Recent message passing kernels (MPGK) combine permutation invariance with efficient, scalable explicit (Nyström) feature approximations, integrating continuous attributes and matching or sum aggregation in a GNN-style recursion (Nikolentzos et al., 2018). For very large datasets, explicit feature vector approaches are preferred for compatibility with linear solvers.

Hybrid and Hierarchical Methods: Graph filtration kernels extend R-convolution by encoding for each feature not just counts but existence intervals over a graph filtration—strictly increasing expressivity over ordinary WL and yielding completeness in certain regimes (Schulz et al., 2021). OT-based kernels leverage geometric information at multiple resolutions, providing positive-definite operators with regularization for computational tractability (Ma et al., 2020).

4. Extensions for Attributes, Geometry, and Context

Attributed Graphs: Many kernels generalize to continuous node- or edge attributes (e.g., GraphHopper, GraphInvariant, Hash-graph, Message Passing GK, RetGK using return probabilities / mean-embedding) (Nikolentzos et al., 2018, Zhang et al., 2018).

Geometric and Topological Graphs: Metric graph kernels via the tropical Torelli map encode the entire geometric structure by mapping the graph to its period (Gram) matrix $\Omega(G)$ and then comparing by a Gaussian or Wasserstein kernel on SPD matrices. These are invariant under edge-refinement and efficiently computable, with strong performance on label-free graph benchmark datasets (Cao et al., 17 May 2025).

Contextualization: Contextual graph kernels extend local substructure counting by annotating each local feature (e.g., subtree) with a concise representation of its context—greatly increasing discriminative power for cases where local motifs alone are insufficient (Navarin et al., 2015).

5. Applications and Empirical Performance

Graph kernels have produced state-of-the-art results in chemoinformatics, bioinformatics, social network analysis, and vision. For example, the Weisfeiler–Lehman (WL) subtree kernel and its variants remain highly competitive on diverse graph-classification datasets (MUTAG, NCI1, PROTEINS), often with accuracy in the 80–89% range (Kriege et al., 2019, Nikolentzos et al., 2019). Optimal-assignment and hybrid WL kernels can further boost accuracy by several points. Spectral, kernel mean-embedding, and density-of-states methods achieve high accuracy on large, attribute-rich graphs (Huang et al., 2020, Zhang et al., 2018).

Semi-structured and geometric kernels (e.g., tropical Torelli) outperform classical motifs on label-free or metric graphs such as urban road networks, where invariance under edge subdivision and sensitivity to global cycles are required (Cao et al., 17 May 2025).

6. Graph Kernels and Deep Learning: Hybridization

Recent trends involve fusing kernel and neural paradigms:

• Message Passing Graph Kernels (MPGK): These model GNN-style neighborhood aggregation but retain interpretability and positive-definiteness by operating entirely in kernel space, offering assignment and R-convolution variants for expressivity (Nikolentzos et al., 2018).
• Graph Kernel Neural Networks / Kernel Graph CNNs: Classical kernels are used as differentiable convolutional operators—either directly on subgraphs (e.g., “mask” matching) or to embed node neighborhoods for convolutional filters in neural nets (Cosmo et al., 2021, Nikolentzos et al., 2017). The resulting architectures combine kernel-driven expressivity with trainability and can match or exceed standard GNN benchmarks on classical datasets.
• Filtration Kernels and Higher-Order GNNs: By using filtrations and additional persistence information, kernels and, by extension, GNN models can surpass the expressivity of the 1-WL test, a known limitation of classical GNN architectures (Schulz et al., 2021).

7. Challenges, Limitations, and Future Directions

• Scalability: Many “complete” or higher-order kernels scale poorly; techniques such as sampling (graphlets, k-WL, quantum superposition), explicit feature maps, and approximation (Count-Sketch, Nyström) are central to practical deployment.
• Expressivity: Most efficient kernels trade off completeness for speed. Higher-dimensional WL, filtration, and context-augmented schemes partially mitigate this.
• Attributed/Geometric Graphs: Extending efficient, expressive kernels to graphs with high-dimensional, continuous, or geometric attributes is an active area—approaches include message-passing, OT, spectral embeddings, and geometric kernel designs (Ma et al., 2020).
• Integration with GNNs: Kernel insights increasingly inform neural architectures (e.g., by kernelizing GNN layers or using kernel features in hybrid pipelines).
• Dynamic and Temporal Graphs: Many static-kernel designs do not naturally extend to time-varying or evolving graph structures; filtration and hierarchical methods are promising in this context.
• Theoretical Hierarchies: There is emerging work comparing expressivity classes of kernels (e.g., filtration > 1-WL, some substructures > others), with implications for both kernel design and neural model limits.

Graph kernels remain an indispensable tool in graph machine learning, offering a balance between theoretical rigor, interpretability, and empirical efficacy. The field is actively evolving toward higher expressivity, scalability, and seamless integration with contemporary deep learning frameworks.

References:

• (Kriege et al., 2019) A Survey on Graph Kernels
• (Nikolentzos et al., 2019) Graph Kernels: A Survey
• (Huang et al., 2020) Density of States Graph Kernels
• (Schulz et al., 2021) Graph Filtration Kernels
• (Nikolentzos et al., 2018) Message Passing Graph Kernels
• (Navarin et al., 2015) Extending local features with contextual information in graph kernels
• (Cao et al., 17 May 2025) Metric Graph Kernels via the Tropical Torelli Map
• (Ma et al., 2020) Transport based Graph Kernels
• (Kishi et al., 2021) Graph kernels encoding features of all subgraphs by quantum superposition
• (Cosmo et al., 2021) Graph Kernel Neural Networks
• (Nikolentzos et al., 2017) Kernel Graph Convolutional Neural Networks
• (0807.0093) Graph Kernels
• (Zhang et al., 2018) RetGK: Graph Kernels based on Return Probabilities of Random Walks