Predicting Instability in Complex Oscillator Networks: Limitations and Potentials of Network Measures and Machine Learning

TL;DR

It is found that no small subset can reliably predict stability and this suggests that correlations of network measures and function may be misleading, and that GNNs capture the causal relationship between structure and stability substantially better.

Abstract

A central question of network science is how functional properties of systems arise from their structure. For networked dynamical systems, structure is typically quantified with network measures. A functional property that is of theoretical and practical interest for oscillatory systems is the stability of synchrony to localized perturbations. Recently, Graph Neural Networks (GNNs) have been shown to predict this stability successfully; at the same time, network measures have struggled to paint a clear picture. Here we collect 46 relevant network measures and find that no small subset can reliably predict stability. The performance of GNNs can only be matched by combining all network measures and nodewise machine learning. However, unlike GNNs, this approach fails to extrapolate from network ensembles to several real power grid topologies. This suggests that correlations of network measures and function may be misleading, and that GNNs capture the causal relationship between structure and stability substantially better.

Figures (15)

• Figure 1: The goal is the prediction of the dynamic stability (targets) based on power grid models (input). Whereas GNNs (at the bottom) deal with the graph input directly, NetSciML models rely on network measures as inputs.
• Figure 2: Basin landscape of a node of a 20-node grid with relatively low stability ($\mathrm{SNBS} \approx 0.67$). The color indicates the maximum absolute frequency deviation of all nodes at the end of the trajectory. Black initial conditions converge back to the synchronous state, others reach desynchronized states. SNBS is equal to the fraction of black points among all 10,000 perturbations.
• Figure 3: Coefficient of determination ($R^2$) of the studied network measures with SNBS for different grid sizes. Global measures of the graph topology, which are the same at every node in a graph, are denoted with [G]. For the Texas grid, global measures are just a single constant, and their $R^2$ is zero. The original source of the features is given in parentheses. Source (NetworkX) indicates that the measure has been added because it was readily available in the widely known Python package hagbergExploringNetworkStructure2008, but has not previously been used for power grid stability to the best of our knowledge.
• Figure 4: Single-node basin stability performance ($R^2$ ) depending on amount of available training data. The plot shows the performance of the best GNN (TAGNet) and the best feature-based model (GBR) for the tasks tr20ev20 (dashed) and tr100ev100 (solid).
• Figure 5: Most important SHAP values of the GBR model for SNBS prediction (evaluated on the test set) of 100 node ensemble (bottom). For the ensemble of 20 nodes, the SHAP values are plotted in \ref{['fig:shap20']}.