Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs
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→−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 H1 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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BibTeX
@misc{forouhesh2026gaugeinvariant,
title = {{Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs}},
author = {Mohammad Forouhesh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.13609},
eprint = {2607.13609},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.13609},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gauge-Invariant, Parameter-Insensitive Regularization for Potential Recovery from Flow on Directed Graphs
AU - Mohammad Forouhesh
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2607.13609
UR - https://arxiv.org/abs/2607.13609
AB - 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→−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.
ER -