Analyzing Neural Network Robustness Using Graph Curvature
Shuhang Tan, Jayson Sia, Paul Bogdan, Radoslav Ivanov
TL;DR
This work reframes neural network robustness as a graph-curvature problem by introducing neural data graphs and neural Ricci curvature ($\mathrm{NRC}$). $\mathrm{NRC}$ is computed as Ollivier-Ricci curvature on graphs constructed from NN edges, with edge weights reflecting both network parameters and input-driven data flow; bottleneck edges correspond to negative curvature and potential robustness vulnerabilities. Empirical results on MNIST show that more $\varepsilon$-robust examples have fewer negative-$\mathrm{NRC}$ edges, suggesting a graph-based path to robustness and a direction for training regularization that minimizes problematic edges. Overall, the paper provides a concrete framework for identifying and potentially mitigating robustness bottlenecks via graph-based curvature, offering an alternative to optimization-heavy robustness methods and guiding future work on multi-dataset evaluation and penalty-based robust training.
Abstract
This paper presents a new look at the neural network (NN) robustness problem, from the point of view of graph theory analysis, specifically graph curvature. Graph curvature (e.g., Ricci curvature) has been used to analyze system dynamics and identify bottlenecks in many domains, including road traffic analysis and internet routing. We define the notion of neural Ricci curvature and use it to identify bottleneck NN edges that are heavily used to ``transport data" to the NN outputs. We provide an evaluation on MNIST that illustrates that such edges indeed occur more frequently for inputs where NNs are less robust. These results will serve as the basis for an alternative method of robust training, by minimizing the number of bottleneck edges.