New Regularization Method Improves Potential Recovery on Directed Graphs

Recovering an underlying "potential" or ranking from observed flow patterns on a directed graph is a fundamental problem in many fields, often framed as a discrete Poisson problem. However, this inverse problem is inherently ill-posed, and conventional solutions like ridge regularization can introduce significant distortions. Specifically, ridge regularization tends to shrink the solution towards an arbitrary origin, leading to a collapse of dynamic range and even a reversal of the true ordering, as demonstrated by a dramatic drop in rank correlation. This research proposes a novel solution by employing gauge-invariant graph Dirichlet energy for regularization. This approach fundamentally resolves the issues associated with ridge regularization, delivering remarkable parameter-insensitivity. The estimated potential remains stable across several orders of magnitude in the regularization parameter, in stark contrast to ridge regularization which can invert the ordering for almost any positive parameter value. The paper provides theoretical proofs that the reduced solve is symmetric positive definite (SPD) and preserves dynamic range, unlike ridge regularization which collapses it. The concept of gauge invariance also extends to graph neural networks (GNNs), where neutralizing the constant mode per layer can prevent oversmoothing, a common problem in deep directed GCNs. This connection links a classical inverse problem to a central challenge in modern graph learning. Empirical tests on three public clickstream datasets show that the gauge-invariant estimate retains a significant portion (28-41%) of the interior dynamic range, whereas ridge regularization collapses it to as little as 0.2%.