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

• URL: http://arxiv.org/abs/2607.13609v1
• Date: Wed, 15 Jul 2026 08:56:57 GMT
• Title: Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs
• Abstract summary: A discrete Poisson problem with Dirichlet boundaries is ill-posed on a directed graph.<n>The standard fix backfires: ridge regularization shrinks toward a gauge-meaningless origin, collapsing and reversing the recovered ordering.<n>We prove the reduced solve is SPD and preserves dynamic range exactly where ridge collapses it, and localize absorbing boundaries from flow alone.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Recovering a latent potential from observed flow on a directed graph (a discrete Poisson problem with Dirichlet boundaries) is ill-posed, and the standard fix backfires: ridge regularization shrinks toward a gauge-meaningless origin, collapsing and reversing the recovered ordering ($+0.81\to-0.42$ rank correlation against a planted ground truth). The gauge-invariant graph Dirichlet energy removes the hazard and delivers parameter-insensitivity: the estimate is stable across four orders of magnitude in $λ$, whereas ridge inverts the ordering for every $λ>0$. We prove the reduced solve is SPD and preserves dynamic range exactly where ridge collapses it, and localize absorbing boundaries from flow alone via a Poisson residual. The $H^1$ seminorm is classical; what is new is the gauge diagnosis, the parameter-insensitivity it buys, and an ablation showing the result is robust to the extraction method. On three public clickstream corpora the gauge-invariant estimate retains $28$--$41\%$ of the interior dynamic range while ridge collapses to as little as $0.2\%$. The same gauge invariance carries into graph neural networks -- neutralizing the constant mode per layer prevents the oversmoothing that collapses a deep directed GCN -- linking this classical inverse problem to a central question in graph learning.

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