Wasserstein-based Graph Alignment

Abstract: We propose a novel method for comparing non-aligned graphs of different sizes, based on the Wasserstein distance between graph signal distributions induced by the respective graph Laplacian matrices. Specifically, we cast a new formulation for the one-to-many graph alignment problem, which aims at matching a node in the smaller graph with one or more nodes in the larger graph. By integrating optimal transport in our graph comparison framework, we generate both a structurally-meaningful graph distance, and a signal transportation plan that models the structure of graph data. The resulting alignment problem is solved with stochastic gradient descent, where we use a novel Dykstra operator to ensure that the solution is a one-to-many (soft) assignment matrix. We demonstrate the performance of our novel framework on graph alignment and graph classification, and we show that our method leads to significant improvements with respect to the state-of-the-art algorithms for each of these tasks.

• The paper introduces a novel framework using optimal transport theory for one-to-many graph alignment.
• It employs a one-to-many soft-assignment strategy with a Dykstra operator and SGD to effectively optimize the alignment process.
• Experimental results show significant improvements in graph alignment recovery and classification accuracy over conventional methods.

Wasserstein-based Graph Alignment

Introduction

The paper "Wasserstein-based Graph Alignment" introduces a novel framework for aligning graphs of disparate sizes using the Wasserstein distance. The primary goal of the paper is to address the challenge of comparing graphs that are not necessarily aligned or of the same size. This is especially relevant in applications like brain connectivity analysis, social network inference, and molecular modeling, where structural alignment presents significant complexities. Traditional graph alignment methods are often NP-hard, making them infeasible for practical use in large datasets.

Methodology

The authors propose a one-to-many graph alignment approach that leverages the Wasserstein distance derived from graph signal distributions. The novel aspect of this methodology is its use of optimal transport theory to define a structurally-meaningful distance measure between graphs.

1. Graph Alignment as Optimal Transport: Utilize the Wasserstein distance between graph signal distributions, which are induced by graph Laplacian matrices. This distance is structurally significant, capturing the intrinsic topology of graph data more effectively than traditional Euclidean or Gromov-Wasserstein distances.
2. One-to-Many Soft-Assignment: The alignment problem is formulated as a one-to-many soft-assignment, where each node in a smaller graph can be aligned to one or more nodes in a larger graph. This is in contrast to previous methods that often assume a one-to-one correspondence.
3. Stochastic Gradient Descent (SGD) and Dykstra Operator: The paper introduces a novel use of stochastic gradient descent to solve the alignment problem efficiently. The Dykstra operator is employed to maintain the integrity of the one-to-many assignment matrix during optimization, ensuring it adheres to the constraints of the problem.

Implementation

The algorithm involves relaxing the one-to-many alignment constraint using a novel Dykstra operator and applying stochastic gradient descent to find optimal alignments. This approach:

• Ensures a soft-assignment matrix that respects the constraints needed for meaningful graph alignment.
• Explores the space of possible solutions more effectively through Bayesian exploration strategies integrated into SGD.

In practice, the algorithm demonstrates strong performance in aligning graphs and detecting community structures within graphs. Implementations can be developed in frameworks like PyTorch, leveraging automatic differentiation to streamline the optimization processes.

Experimental Results

The effectiveness of the proposed method is illustrated through its application in both graph alignment and classification tasks on synthetic and real datasets. Numerical results show significant improvements in graph alignment and classification accuracy compared to state-of-the-art methods like Gromov-Wasserstein distances and conventional $\ell_2$-norm distances.

1. Graph Alignment: The algorithm exhibits superior performance in recovering community structures and achieving alignment with lower distortion, especially amid significant structural perturbations.
2. Graph Classification: On datasets like PTC and IMDB-B, the Wasserstein-based approach improves accuracy over several state-of-the-art baseline methods, including those using Euclidean and Gromov-Wasserstein metrics.

Conclusion

This paper contributes a significant advancement in graph alignment by framing the problem in the context of optimal transport and introducing mechanisms, like the Dykstra operator, that facilitate effective optimization solutions. While demonstrating improvements over existing methods, the approach also highlights several areas for future work, including scalability improvements and refinements to handle even larger and more complex datasets with greater computational efficiency. This fills a critical gap in the domain of graph-based data analysis, providing a robust tool for tackling previously intractable graph comparison problems.