Enhancing Robustness of Graph Neural Networks through p-Laplacian

TL;DR

The paper addresses the vulnerability of Graph Neural Networks to adversarial perturbations by introducing pLAPGNN, a two-stage framework that denoises the graph topology using a weighted $p$-Laplacian and then trains a robust GNN on the cleaned structure. It combines a nonlinear $p$-Laplacian regularizer with a noise-removal objective and a standard GNN loss, solved via a Majorization-Minimization scheme that first reconstructs an accurate Laplacian $\Phi^*$ from the noisy $\Phi_n$ and then optimizes the GNN parameters on the denoised graph. Empirical results on Cora, Citeseer, and PubMed under Nettack and Metattack show that $p$LAPGNN consistently improves robustness, converges faster, and reduces computational overhead relative to SVD-based defenses, thanks to its adaptive regularization and decoupled denoising-and-learning stages. The work suggests that nonlinear $p$-Laplacian regularization offers a flexible, scalable path to robust graph learning, with potential extensions to larger, heterogeneous, and dynamic graphs.

Abstract

With the increase of data in day-to-day life, businesses and different stakeholders need to analyze the data for better predictions. Traditionally, relational data has been a source of various insights, but with the increase in computational power and the need to understand deeper relationships between entities, the need to design new techniques has arisen. For this graph data analysis has become an extraordinary tool for understanding the data, which reveals more realistic and flexible modelling of complex relationships. Recently, Graph Neural Networks (GNNs) have shown great promise in various applications, such as social network analysis, recommendation systems, drug discovery, and more. However, many adversarial attacks can happen over the data, whether during training (poisoning attack) or during testing (evasion attack), which can adversely manipulate the desired outcome from the GNN model. Therefore, it is crucial to make the GNNs robust to such attacks. The existing robustness methods are computationally demanding and perform poorly when the intensity of attack increases. This paper presents a computationally efficient framework, namely, pLAPGNN, based on weighted p-Laplacian for making GNNs robust. Empirical evaluation on real datasets establishes the efficacy and efficiency of the proposed method.

Figures (5)

• Figure 1: Normalized Dirichlet energy under increasing adversarial perturbations on different datasets. A consistent rise indicates a loss of feature smoothness across edges, motivating the need for structure-preserving defenses such as $p$LapGNN.
• Figure 2: Proposed framework. Dashed lines indicate adversarial edges incorporated during the optimization process.
• Figure 3: Performance of various models under Nettack on three benchmark datasets. Our proposed method $p$LapGNN maintains high robustness across all perturbation levels.
• Figure 4: Effect of the $p$ parameter on robustness (Cora, 10% Meta-attack). Accuracy peaks at $p=2.4$, indicating a balance between sparsity ($p < 2$) and stability ($p > 2$). Error bars denote standard deviations over five runs.
• Figure : $p$LapGNN: MM Denoising and GNN Training