The Multiscale Laplacian Graph Kernel (1603.06186v2)

Abstract: Many real world graphs, such as the graphs of molecules, exhibit structure at multiple different scales, but most existing kernels between graphs are either purely local or purely global in character. In contrast, by building a hierarchy of nested subgraphs, the Multiscale Laplacian Graph kernels (MLG kernels) that we define in this paper can account for structure at a range of different scales. At the heart of the MLG construction is another new graph kernel, called the Feature Space Laplacian Graph kernel (FLG kernel), which has the property that it can lift a base kernel defined on the vertices of two graphs to a kernel between the graphs. The MLG kernel applies such FLG kernels to subgraphs recursively. To make the MLG kernel computationally feasible, we also introduce a randomized projection procedure, similar to the Nystr\"om method, but for RKHS operators.

• The paper introduces the FLG and MLG kernels that capture graph similarities across scales by combining local node features with global structure.
• It develops a recursive kernel construction and randomized low-rank approximation to reduce computational complexity for large graphs.
• Numerical experiments show superior classification accuracy on benchmark datasets like MUTAG, PTC, and ENZYMES, indicating broad applications.

The Multiscale Laplacian Graph Kernel: A Comprehensive Overview

This paper introduces two novel graph kernels that address the challenge of capturing structural similarities across multiple scales in graphs, a common occurrence in domains such as chemoinformatics and social network analysis. Existing graph kernels typically focus on either local or global similarities, which limits their applicability in scenarios where both aspects are crucial. The proposed Multiscale Laplacian Graph (MLG) kernel provides a mechanism to capture these multiscale structures effectively.

Key Contributions and Methodology

The main contributions of this paper are the introduction of the Feature Space Laplacian Graph (FLG) kernel and the Multiscale Laplacian Graph (MLG) kernel. The approach involves several innovative steps:

1. Feature Space Laplacian Graph (FLG) Kernel: This novel kernel leverages a combination of node features and the graph Laplacian to assess similarity between graphs. The FLG kernel constructs a covariance matrix over vertex features, allowing it to encapsulate both local node information and overall graph structure. Importantly, it lifts a base kernel defined on graph vertices to a higher-level kernel between graphs.
2. Recursive Kernel Construction: To allow for efficient computation over large graphs, a recursive method is devised. The MLG kernel uses the FLG kernel to recursively build a hierarchy of kernels, enabling the analysis of subgraphs of increasing sizes. This recursive kernel is particularly adept at capturing intricate relationships between subgraphs and their topological contexts within the overall graph.
3. Randomized Low-Rank Approximation: The authors propose a randomized projection procedure akin to the Nyström method to make kernel computations feasible for large datasets. This approach approximates the eigenspectrum of graph Laplacians, thereby reducing computational complexity from prohibitive levels for large graphs.

Numerical Experiments and Results

The empirical evaluation of the MLG kernel demonstrates its superiority in terms of classification accuracy over other advanced methods, such as the Weisfeiler–Lehman kernel and various spectral and random walk kernels. On benchmark datasets like MUTAG, PTC, and ENZYMES, the MLG kernel delivers enhanced performance, particularly in capturing the nuanced structural similarities that are inadequately handled by traditional graph kernels focusing on single-scale graph properties.

Implications and Future Directions

The MLG kernel represents a significant step forward in graph analysis by enabling multiscale similarity assessments. While the initial focus is on chemoinformatics and social network analysis, the recursive kernel construction strategy can be beneficial across various domains where multiscale data plays a critical role, such as biomedical image analysis and network biology.

Looking ahead, the methods introduced in this paper could be refined with domain-specific adaptations to further enhance performance. The integration of advanced machine learning techniques, such as deep learning on graphs, may also provide additional avenues for leveraging the multiscale capabilities of the proposed kernels. Furthermore, investigating the application of these kernels to dynamic graphs could open new research paths in evolving network analysis.

In conclusion, the Multiscale Laplacian Graph kernel framework provides a robust and flexible approach to capturing complex multiscale relationships in graph data. By addressing both computational efficiency and structural expressiveness, this methodological advancement holds significant promise for numerous applications in both theoretical and applied graph analysis.