GRASP: Graph Alignment through Spectral Signatures (2106.05729v2)

Abstract: What is the best way to match the nodes of two graphs? This graph alignment problem generalizes graph isomorphism and arises in applications from social network analysis to bioinformatics. Some solutions assume that auxiliary information on known matches or node or edge attributes is available, or utilize arbitrary graph features. Such methods fare poorly in the pure form of the problem, in which only graph structures are given. Other proposals translate the problem to one of aligning node embeddings, yet, by doing so, provide only a single-scale view of the graph. In this paper, we transfer the shape-analysis concept of functional maps from the continuous to the discrete case, and treat the graph alignment problem as a special case of the problem of finding a mapping between functions on graphs. We present GRASP, a method that first establishes a correspondence between functions derived from Laplacian matrix eigenvectors, which capture multiscale structural characteristics, and then exploits this correspondence to align nodes. Our experimental study, featuring noise levels higher than anything used in previous studies, shows that GRASP outperforms state-of-the-art methods for graph alignment across noise levels and graph types.

• The paper introduces GRASP, a robust graph alignment method that leverages spectral signatures derived from Laplacian eigenvectors to match nodes without external attributes.
• The methodology employs heat kernel analysis and eigenvector alignment to transform graph alignment into a functional mapping between different eigenspaces.
• Empirical evaluation demonstrates that GRASP outperforms state-of-the-art techniques in noisy conditions across applications like social network analysis and bioinformatics.

Overview

The paper, "GRASP: Graph Alignment through Spectral Signatures" explores a novel method for aligning nodes in graphs by leveraging spectral signatures derived from the graphs' Laplacian matrices. The goal is to find a correspondence between nodes of two graphs based solely on their structural configurations without requiring additional information like node attributes or known seed matches. This work is significant for applications across fields such as social network analysis and bioinformatics, where such pure graph alignment problems frequently arise.

Methodology

Functional Mapping and Spectral Signatures

The core concept of GRASP is to treat the graph alignment problem as a mapping between functions on graphs, drawing inspiration from shape analysis techniques. Specifically, GRASP utilizes the Laplacian matrix eigenvectors to capture multiscale structural characteristics of graphs. These eigenvectors act as spectral signatures that allow the encoding of both local and global graph structures.

Heat Kernel

The method employs the heat kernel of the graph, which describes heat diffusion over the graph's structure at various time scales. This provides a multiscale representation of the graph's structure. The diagonal of the heat kernel is used as a signature for each node. This description harnesses the expressive power of graph spectra, which capture information ranging from local edge connectivity to long-range community structures.

Mapping Matrix and Alignment

The approach uses a linear mapping between these spectral signatures of the respective graphs to construct a mapping matrix $C$. The whole process is framed as a functional map between different eigenspaces to capture node correspondences based on the spectral signatures.

An assignment problem is then solved, aligning the nodes by matching the computed spectral signatures. Techniques such as nearest neighbor searches or the Jonker-Volgenant (JV) algorithm are considered for efficient computation of these alignments.

Implementation Details

Computational Steps

1. Eigenvector Calculation: The first step involves calculating the eigenvectors and eigenvalues of the normalized Laplacian matrices of the input graphs. Only a subset of $k$ eigenvectors is considered to manage complexity.
2. Signature Computation: For a series of time steps, the heat kernel diagonal is computed from the eigenvectors. This heat kernel serves as the node signature at various resolutions.
3. Base Alignment: The eigenvector bases are aligned using a rotation matrix $M$ to account for differences in the structure of the eigenvector spaces between the graphs.
4. Assignment Problem: The final node alignment is conducted using the calculated mapping of spectral signatures.

Computational Complexity

The method is designed for scalability by focusing on a reduced set of leading eigenvectors, thus avoiding the full eigendecomposition potentially required for large graphs. The use of spectral properties facilitates efficient computation relative to the graph's size.

Empirical Evaluation

The paper provides empirical evidence showing that the GRASP method outperforms state-of-the-art methods in accuracy when tested across various types of noisy graph scenarios. The experiments demonstrated that GRASP's utilization of multiscale spectral properties enables it to maintain high alignment performance even in challenging conditions with higher noise levels.

Implications and Future Directions

GRASP's approach has broad implications, offering a more robust solution to graph alignment problems by leveraging deep structural insights provided by spectral analysis. This technique is potentially powerful for analyzing large-scale networks in various domains, including computational biology and social network mining.

Future extensions could explore the integration of partial correspondence solutions and expand GRASP's applicability to more varied types of graph data and alignment scenarios. The method's framework could further benefit from incorporating machine learning techniques to refine node matching and improve computational efficiency through strategic approximations.