Stability Properties of Graph Neural Networks (1905.04497v5)

Abstract: Graph neural networks (GNNs) have emerged as a powerful tool for nonlinear processing of graph signals, exhibiting success in recommender systems, power outage prediction, and motion planning, among others. GNNs consists of a cascade of layers, each of which applies a graph convolution, followed by a pointwise nonlinearity. In this work, we study the impact that changes in the underlying topology have on the output of the GNN. First, we show that GNNs are permutation equivariant, which implies that they effectively exploit internal symmetries of the underlying topology. Then, we prove that graph convolutions with integral Lipschitz filters, in combination with the frequency mixing effect of the corresponding nonlinearities, yields an architecture that is both stable to small changes in the underlying topology and discriminative of information located at high frequencies. These are two properties that cannot simultaneously hold when using only linear graph filters, which are either discriminative or stable, thus explaining the superior performance of GNNs.

• The paper establishes that employing integral Lipschitz filters ensures GNN stability under both absolute and relative perturbation models.
• The paper uses graph spectral theory and permutation equivariance to analyze GNN sensitivity to noise and adversarial attacks.
• The paper reveals that integrating pointwise nonlinearities like ReLU enhances information mixing, preserving robust inference despite graph perturbations.

Stability Properties of Graph Neural Networks

The paper under review addresses the stability of Graph Neural Networks (GNNs) in response to changes in the underlying graph topology. The significance of this topic lies in the intrinsic variability of real-world networks where GNNs are applied, such as social networks, sensor networks, and biological networks. Small perturbations in the topology may arise from measurement noise, communication errors, or adversarial attacks, raising the importance of understanding how such variations impact the performance of GNNs.

Main Contributions and Findings

The authors contribute to the field by providing a theoretical analysis of the GNNs' stability, building on the concept of graph signal processing. The paper leverages tools such as graph spectral theory and the notion of permutation equivariance to investigate GNN behavior under perturbations. Several key contributions and findings are presented:

1. Permutation Equivariance: The paper illustrates that GNNs inherit permutation equivariance from graph filters. This property ensures that the node labeling or arbitrary relabeling in the graph does not affect the output of the GNN. This is significant as it guarantees that GNNs can consistently exploit the intrinsic symmetries within the graph data.
2. Stability with Respect to Graph Perturbations: The paper delineates two models of perturbation: absolute and relative. The absolute perturbation model considers fixed changes to graph edges, whereas the relative perturbation model ties the perturbations to the local structure of the graph, making it more realistic for many practical applications.
3. Lipschitz vs. Integral Lipschitz Filters: The paper explores the frequency response of graph filters, distinguishing between Lipschitz filters and integral Lipschitz filters. The latter impose a stricter rate of change near high frequencies, which is crucial for ensuring stability under the relative perturbation model. The paper shows that Lipschitz filters can be sensitive to changes at high frequencies, whereas integral Lipschitz filters mitigate this sensitivity by ensuring a near-constant response at these frequencies.
4. Nonlinearities in GNNs: It is posited that the integration of pointwise nonlinearities, such as ReLU, within GNNs facilitates information mixing over the spectrum. This mixing allows low-frequency stable filters to capture information initially located at high-frequency components, enhancing the discriminative power of GNNs without sacrificing stability.
5. Theoretical Implications: It was demonstrated that GNN architectures can be stable under small perturbations if integral Lipschitz filters are employed. This model implies that GNNs can maintain high classification or inference accuracy even if the input graph is inaccurately estimated or subject to noise.

Implications and Speculations

The findings in this paper have profound implications for both the theoretical understanding and practical deployment of GNNs:

• The theoretical guarantees on stability provide a foundational understanding that could drive the development of more robust GNN architectures. This can encourage further exploration into designing customized filters or architectures tailored to specific types of graph transformations or noise.
• In practice, enforcing or promoting integral Lipschitz constraints during the training of GNNs could lead to models that are innately more robust to perturbations in real-world scenarios.
• The methodology outlined could foster advancements in areas that require high resilience to graph changes, such as cybersecurity (defending against network attacks), dynamic network analysis, and adaptive sensor networks.
• Future research could extend stability guarantees to other deep learning architectures on graphs, including non-Convolutional architectures, or explore the transferability of these stability properties across different datasets or domains.

This paper advances the understanding of the structural properties of GNNs, delineating the strong interplay between graph filters' spectral properties, neural network nonlinearities, and model robustness. This contribution is expected to guide both theorists and practitioners in the escalating endeavor to deploy GNNs effectively and reliably in diverse applications.