Facing climate change, the transformation to renewable energy poses stability challenges for power grids due to their reduced inertia and increased decentralization. Traditional dynamic stability assessments, crucial for safe grid operation with higher renewable shares, are computationally expensive and unsuitable for large-scale grids in the real world. Although multiple proofs in the network science have shown that network measures, which quantify the structural characteristics of networked dynamical systems, have the potential to facilitate basin stability prediction, no studies to date have demonstrated their ability to efficiently generalize to real-world grids. With recent breakthroughs in Graph Neural Networks (GNNs), we are surprised to find that there is still a lack of a common foundation about: Whether network measures can enhance GNNs' capability to predict dynamic stability and how they might help GNNs generalize to realistic grid topologies. In this paper, we conduct, for the first time, a comprehensive analysis of 48 network measures in GNN-based stability assessments, introducing two strategies for their integration into the GNN framework. We uncover that prioritizing measures with consistent distributions across different grids as the input or regarding measures as auxiliary supervised information improves the model's generalization ability to realistic grid topologies, even when models trained on only 20-node synthetic datasets are used. Our empirical results demonstrate a significant enhancement in model generalizability, increasing the$R^{2}$perforsmance from 66% to 83%. When evaluating the probabilistic stability indices on the realistic Texan grid model, GNNs reduce the time needed from 28,950 hours (Monte Carlo sampling) to just 0.06 seconds.

Diffusion probabilistic models (DPMs) have recently demonstrated impressive generative capabilities. There is emerging evidence that their sample reconstruction ability can yield meaningful representations for recognition tasks. In this paper, we demonstrate that the objectives underlying generation and representation learning are not perfectly aligned. Through a spectral analysis, we find that minimizing the mean squared error (MSE) between the original graph and its reconstructed counterpart does not necessarily optimize representations for downstream tasks. Instead, focusing on reconstructing a small subset of features, specifically those capturing global information, proves to be more effective for learning powerful representations. Motivated by these insights, we propose a novel framework, the Smooth Diffusion Model for Graphs (SDMG), which introduces a multi-scale smoothing loss and low-frequency information encoders to promote the recovery of global, low-frequency details, while suppressing irrelevant high-frequency noise. Extensive experiments validate the effectiveness of our method, suggesting a promising direction for advancing diffusion models in graph representation learning.