Interpreting Temporal Graph Neural Networks with Koopman Theory (2410.13469v1)

Abstract: Spatiotemporal graph neural networks (STGNNs) have shown promising results in many domains, from forecasting to epidemiology. However, understanding the dynamics learned by these models and explaining their behaviour is significantly more complex than for models dealing with static data. Inspired by Koopman theory, which allows a simpler description of intricate, nonlinear dynamical systems, we introduce an explainability approach for temporal graphs. We present two methods to interpret the STGNN's decision process and identify the most relevant spatial and temporal patterns in the input for the task at hand. The first relies on dynamic mode decomposition (DMD), a Koopman-inspired dimensionality reduction method. The second relies on sparse identification of nonlinear dynamics (SINDy), a popular method for discovering governing equations, which we use for the first time as a general tool for explainability. We show how our methods can correctly identify interpretable features such as infection times and infected nodes in the context of dissemination processes.

• The paper introduces a novel explainability framework that leverages Koopman theory with DMD and SINDy to transform STGNN dynamics into interpretable linear models.
• The methodology accurately identifies crucial temporal changes via F1 scores and reveals spatial dependencies with strong AUC performance across diverse datasets.
• The findings improve the interpretability of temporal graph models, paving the way for real-time insights and broader applications in dynamic network analysis.

Interpreting Temporal Graph Neural Networks with Koopman Theory

This paper introduces a novel explainability framework for spatiotemporal graph neural networks (STGNNs) utilizing Koopman theory. Temporal Graphs (TGs) are integral in modeling complex spatiotemporal relationships found in various domains, including traffic dynamics, social interactions, and epidemiology. While STGNNs have shown efficacy in these applications, their interpretability remains a significant challenge. This work aims to address this through an innovative application of Koopman theory, which traditionally transforms nonlinear dynamical systems into linear representations, thus aiding in the interpretation of their behaviors.

Methodology

The authors propose two distinct methods to interpret the decision-making process of STGNNs based on Koopman-inspired techniques:

1. Dynamic Mode Decomposition (DMD): This method is used to identify key temporal and spatial patterns in the input data by approximating the Koopman operator. DMD has been adapted from fluid dynamics to work with the embeddings generated by STGNNs, allowing the detection of significant temporal changes in the data.
2. Sparse Identification of Nonlinear Dynamics (SINDy): Newly adapted for explainability in this context, SINDy helps in discovering governing equations from data. This method is employed to elucidate the relevance of specific spatial relationships within the graph, particularly inter-node dependencies, which are critical for accurate model predictions.

The model trains a Graph Convolutional Recurrent Network (GCRN) to transform the complex dynamics of input TGs into a linear dynamic system in the embedding space. This is achieved using a novel internal state and regularization techniques that incorporate Koopman theory, thus ensuring that the latent space dynamics remain interpretable while preserving task-related performance.

Key Findings

Experiments were conducted on TGs derived from various domains, including social interactions and dissemination processes. The interpretation methods successfully highlighted relevant spatial and temporal features such as infection times and the nodes involved. The models achieved high classification accuracy, with results demonstrating:

• Temporal explanation accuracy as evaluated by F1 scores and Mann-Whitney U tests indicating a reliable identification of significant time steps in dynamical events.
• Spatial explanation efficiency highlighted by Area Under the Curve (AUC) scores where both node-level and edge-level explanations correlated well with the infection ground truths.

Implications

The application of Koopman theory to STGNNs presents a significant stride towards making these powerful models more interpretable without sacrificing accuracy. This framework provides a way to understand complex temporal and spatial dependencies directly from data, which could be beneficial for practitioners dealing with dynamic networks and systems. Moreover, the successful integration of SINDy into this context opens up further exploration into discovering governing dynamics in other complex systems beyond TGs.

Future Directions

This research lays the groundwork for several potential future developments:

• Broader Application Scope: Extending these interpretability methods to other graph neural networks and different types of temporal data could generalize insights beyond current applications.
• Exploration of Nonlinear Dynamics: Further integration with nonlinear dynamical systems could enhance the understanding and robustness of STGNN models.
• Real-Time Interpretability: Developing methods to apply these interpretability techniques in real-time could provide immediate insights during dynamic processes.

Overall, this paper enriches the interpretability toolbox for temporal graph models, facilitating better adoption and trust in machine learning models applied to complex dynamic data.