Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs

Recovering a latent scalar potential from observed directed flow is a classic inverse problem, analogous to solving a Poisson equation backwards. In many real-world scenarios—such as traffic networks, supply chains, or navigation logs—the flow data is noisy and the graph structure is complex. Standard approaches typically use Tikhonov regularization (ridge regression) to stabilize the ill-posed system. However, this paper identifies a critical failure: because the potential is only defined up to an additive constant (a gauge), ridge regularization forces the solution toward an arbitrary zero-point. This creates a bias that does not just reduce precision, but actively inverts the rank ordering of the nodes, turning a meaningful signal into a misleading one.

The author proposes replacing the standard magnitude penalty with a graph-Sobolev penalty, specifically the graph Dirichlet energy. Unlike ridge, which penalizes the amplitude of the potential, the Dirichlet energy penalizes the differences between connected nodes. Because this penalty is flat along the constant mode, it is gauge-invariant. This shift transforms the regularization from a delicate tuning problem—where the wrong choice of lambda destroys the data—into a robust, parameter-insensitive process. The estimate remains stable across four orders of magnitude, preserving the dynamic range and the correct ranking of nodes even when the regularization strength varies significantly.

This work bridges the gap between classical inverse problems and modern graph learning. The author shows that the same gauge-invariance principle can prevent oversmoothing in deep directed graph neural networks. By neutralizing the constant mode at each layer, the model avoids the common failure mode where node representations collapse into a single value. This provides a unified, theoretically grounded approach to both recovering interpretable scalar potentials and improving the stability of deep learning architectures on directed graphs.