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

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

Mohammad Forouhesh Amirkabir University of Technology, Iran [email protected]

arXiv:2607.13609v1 [cs.LG] 15 Jul 2026

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 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.

1

Introduction

Regularizing an ill-posed inverse problem on a graph requires two decisions: which penalty to add, and how strongly. They are usually treated separately, on the assumption that a reasonable penalty degrades gracefully as λ varies. This paper concerns a setting where that fails: the conventional penalty does not degrade gracefully but inverts the solution, so λ becomes a choice between a usable answer and a confidently wrong one. The setting is recovery of a latent scalar potential from observed directed flow—a traffic network through flow counts, a supply chain through inventory movement, a navigation log through transition frequencies: the discrete analogue of solving a Poisson equation backwards from divergence to potential. We identify when this happens, explain why, and show a different penalty removes it, leaving the answer insensitive to λ across orders of magnitude. Setup and degeneracy. Let G = (V, E, W ) be a weighted directed acyclic graph with absorbing boundary B ⊂ V . A potential ϕ induces the flux qij = Wij (ϕj − ϕi ),

(1) P

P

the discrete analogue of Ohm’s and Fick’s laws, with net divergence bi = j qji − j qij . Substituting gives the directed discrete Poisson equation Lϕ = b with L the weighted Laplacian. It recovers ϕ from a noisy empirical b subject to Dirichlet values on B, and is ill-posed (L1 = 0, low-divergence chains, finite data), so regularization is unavoidable; the standard choice is general-form Tikhonov, 2 2 arg minϕ 12 ∥Lϕ − b∥2 + λ ∥Rϕ∥2 , with the default ridge R = I. The gauge mismatch. The potential is fixed only up to an additive constant, set when boundary values are assigned. Ridge shrinks toward 0, but 0 is not distinguished in a Dirichlet problem—it is wherever the boundary placed the origin. On a positive-boundary graph the pull is asymmetric, Preprint. Preliminary work.

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

dragging interior states toward the abandon boundary regardless of their true position, collapsing the range and, past a small λ, reversing the order. On a controlled instrument with planted ground truth (Section 7), the ridge rank correlation falls from +0.81 at λ = 0 to ≈ − 0.42, inverting the ordering, with linear correlation crossing zero (Figure 1); the only safe ridge setting is λ → 0—no regularization. Parameter-insensitivity. The fix is to make the penalty blind to the gauge P by taking R to be the incidence operator, so the penalty is the graph Dirichlet energy ϕ⊤ LG ϕ = (u,v)∈E wuv (ϕv − ϕu )2 (the graph H 1 seminorm); we call the resulting estimator graph-Sobolev regularization. Because LG 1 = 0, this penalizes differences, not amplitude, and is flat along the gauge mode. The consequence is practical: the estimate is origin-invariant and stable in λ, holding rank correlation at +0.81 and linear correlation in [+0.76, +0.85] across λ ∈ [10−3 , 10], so the strength cannot be misset. Where ridge forces a knife-edge choice, the Dirichlet energy removes it. What is, and is not, new. The penalty itself is old: the H 1 seminorm is the classical alternative to the identity seminorm (general-form vs. standard-form Tikhonov (Hansen, 1998; Tikhonov and Arsenin, 1977)), and ϕ⊤ Lϕ is the standard smoothness functional in graph signal processing and semi-supervised learning (Shuman et al., 2013; Zhou et al., 2004; Belkin and Niyogi, 2003; Zhu et al., 2003). What is new is: (i) those priors interpolate a partially observed node signal, whereas ours is an inhomogeneous inverse problem deconvolving a potential from its divergence with Dirichlet boundaries, where the decisive property is gauge invariance, not smoothness; (ii) ridge is not merely suboptimal but actively harmful, inverting the ordering, with the gauge diagnosis explaining why; (iii) the parameter-insensitivity this yields; (iv) a Poisson-residual boundary diagnostic; and (v) a behavioral-flow application with a planted-ground-truth instrument. Edge-preserving gauge-invariant penalties (total variation, the p-Laplacian (Rudin et al., 1992)) are discussed in Section 2. Contributions. (1) Parameter-insensitivity: the gauge-invariant penalty makes the recovered potential stable across four orders of magnitude in λ (Section 5); (2) the ridge inversion and its gauge diagnosis (+0.81 → −0.42 rank correlation); (3) discrete guarantees—gauge invariance (Proposition 5), an SPD reduced system that scales to 104 nodes by conjugate gradients (Theorem 1), and exact range preservation on chains (Theorem 2); and (4) a reproducible pipeline1 with a Poisson-residual boundary diagnostic (Section 6), an extraction ablation, real-corpus validation, and a downstream task where the gauge-invariant potential is a usable node feature and ridge’s is not (Sections 7 and 8). We are precise about scope: the penalty guarantees preservation, not signal—whatever ordering the data supports is preserved across λ, not destroyed.

2

Related Work

Inverse problems and flows on graphs. Recovering a potential from observed flow is the discrete counterpart of an elliptic inverse problem. The electrical-network analogy is classical: a resistor network with prescribed boundary potentials obeys a discrete Laplace equation whose harmonic functions encode absorption probabilities of the associated walk (Doyle and Snell, 1984; Grady, 2006). Closest to our setting, Jia et al. (2019) recover edge flows by Hodge-based semi-supervised learning, and Schaub and Segarra (2018) denoise edge flows by projecting onto Hodge gradient/curl subspaces. We differ in recovering the node potential through the directed forward operator with Dirichlet boundaries, and in diagnosing the gauge-induced inversion that the identity seminorm causes, a failure mode about the regularizer, not the representation. Regularization and graph signal processing. Tikhonov regularization is the default stabilizer for ill-posed systems (Tikhonov and Arsenin, 1977; Hansen, 1998), the seminorm matrix encoding prior structure; total-variation and graph trend-filtering penalties trade quadratic smoothness for sharp transitions (Rudin et al., 1992; Wang et al., 2016). Whether the penalty seminorm annihilates the forward operator’s null space governs the estimator’s bias. Known in principle, we make it concrete and consequential for the directed Dirichlet problem: the identity seminorm does not annihilate the gauge mode, and the resulting bias is not lost resolution but a reversal of order. The same quadratic form ϕ⊤ LG ϕ is the standard smoothness functional in graph signal processing and Laplacian semisupervised learning (Shuman et al., 2013; Belkin and Niyogi, 2003; Zhu et al., 2003; Zhou et al., 1

Code: https://github.com/MohammadForouhesh/gauge-flow-recovery 2

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

2004), the “trivial” structural choice that sheaf and connection Laplacians generalize (Bodnar et al., 2022)—but there for a signal partially observed on the nodes, not, as here, an inhomogeneous inverse problem where gauge invariance is decisive. Directed graphs and the Hodge decomposition. Directed graph neural networks encode direction in a complex Hermitian (magnetic) Laplacian (Zhang et al., 2021; He et al., 2022) or directionalityaware message passing (Rossi et al., 2023), and directed graph signal processing studies edge directionality (Marques et al., 2020); these target node/edge representation learning, whereas we keep a real directed Laplacian as a forward operator and regularize its inverse. The Helmholtz–Hodge decomposition (Jiang et al., 2011; Lim, 2020; Schaub et al., 2020) splits an edge flow into gradient, curl, and harmonic parts; we use the gradient only to orient flow into an acyclic support, which Sections 7 and 7.1 show is not load-bearing. Graph total variation, trend filtering, and the p-Laplacian (Rudin et al., 1992; Wang et al., 2016) are gauge-invariant alternatives; the dividing line for the ridge pathology is gauge invariance, which they share and the identity seminorm lacks, not the exponent. Connection to graph representation learning. Operationally the gauge-invariant potential is a directed-graph node feature, useful downstream where the ridge potential is not (Section 8.4). Its collapse is an instance of oversmoothing —repeated propagation driving node representations toward the constant mode (Bodnar et al., 2022), just as the magnitude penalty collapses the potential’s range; we verify it (Table 6), a vanilla directed GCN’s conversion AUC falling from 0.86 (2 layers) to 0.81 (32) as its node energy collapses. The same gauge invariance fixes it: neutralizing the constant mode at each layer—the per-layer analogue of LG 1 = 0—holds AUC flat (0.85 at 32 layers), recovering PairNorm and Dirichlet-energy-constrained networks (Zhao and Akoglu, 2020; Zhou et al., 2021) from one principle: flatness along the gauge mode both stabilizes the inverse problem and prevents oversmoothing. Where magnetic- and sheaf-Laplacian networks (Zhang et al., 2021; Bodnar et al., 2022) learn embeddings, we recover one interpretable scalar per node through an explicit forward operator.

3

From Raw Flows to an Acyclic Support

The inverse problem is posed on a directed acyclic graph, but raw flow is neither acyclic nor purely gradient: it carries reciprocal traffic, transient loops, and a solenoidal component no potential can explain. Extraction is upstream machinery; Section 7.1 shows the paper’s claims are invariant to how it is done, so we keep the description brief and defer mechanics to Section A. Flow and dominance. Sessions are state sequences; collapsing consecutive repeats and counting transitions gives the empirical flow Fuv , used as the conductance Wuv = Fuv . Reciprocal traffic is filtered by a dominance test: for ρ ≥ 1, the pair (u, v) is dominant if Fuv ≥ ρFvu . Orientation. An acyclic support needs an orientation of the dominant edges. When session order is available, the default is a cheap topological sort: order states by mean visit position (or net inflow), keeping dominant edges that respect the order. This needs no linear solve, and Section 7.1 shows it recovers the potential at least as well as the alternatives. When only an aggregate flow matrix is available (no session order to sort by), we instead orient by the discrete Helmholtz–Hodge projection: the gradient potential ϕ0 solving G⊤ G ϕ0 = G⊤ ω for the skew flow ωuv = Fuv − Fvu , with edges oriented by increasing ϕ0 (Jia et al., 2019; Schaub and Segarra, 2018). It needs only the flow, but the ablation finds it no more accurate than the topological sort and slower; we keep it solely for the order-free setting. Retention rules for both are in Section A. Acyclicity and the Poisson right-hand side. Proposition 1 (Acyclicity). Either orientation induces a topological order of the retained support Gδ = (V, Eδ ); hence Gδ is acyclic. Let A be the substochastic transition operator on Gδ (outgoing weights normalized, absorbing rows at sinks). Since Gδ is a DAG, A is nilpotent. Proposition 2 (Nilpotency). If the longest directed path in Gδ has length L, then Am = 0 for some m ≤ L + 1. 3

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

Nilpotency is the algebraic certificate that extraction succeeded: a single residual cycle would make m Am ̸= 0 for all m; we verify P ∥A ∥FP= 0 to machine precision on the instrument (Section 7). The empirical divergence bi = j Fji − j Fij is the Poisson right-hand side (Section 4). Propositions 1 and 2 are proved in Section B.

4

The Poisson Inverse Problem

Constitutive law. The flux law Eq. (1) is the optimality condition of a minimum-dissipation principle, pinning down the operator the regularizer must respect. Proposition 3 (Minimum P dissipation). 2Among all edge flows q with prescribed divergence div(q) = b, the one minimizing 12 (i,j)∈E Wij−1 qij is a gradient flow: there exists ϕ with qij = Wij (ϕj − ϕi ), and ϕ solves Lϕ = b. Least-squares objective and its degeneracies. Given a noisy empirical b, the natural estimator minimizes the flow residual 2

E(ϕ) = 12 ∥Lϕ − b∥2 ,

L⊤ L ϕ = L⊤ b.

(2)

Two degeneracies make Eq. (2) ill-posed even on exact b: L1 = 0 leaves ϕ determined only up to an additive gauge, and interior chains carry no divergence, which flattens the solution. Lemma 1 (Chain saturation). On an interior chain v1 → · · · → vk → s with bvi = 0 for every interior vi and ϕ(s) pinned, any solution of Lϕ = b satisfies ϕ(v1 ) = · · · = ϕ(vk ) = ϕ(s). Lemma 1 is why the regularizer choice matters: a chain encodes the ordering of its states, but the unregularized solve flattens it. A penalty that further compresses the chain (as ridge does) erases the ordering; one biasing toward a smooth ramp recovers it—the distinction Section 5 makes precise. Dirichlet boundaries fix the gauge. We resolve the constant-mode degeneracy by pinning the potential on the absorbing boundary B: conversion sinks at 1 and an abandon sink at 0. Partitioning V = I ∪ B and writing L, b in block form, the reduced unknown ϕI has effective right-hand side bI − LIB ϕB and operator L⊤ II LII . This removes the gauge freedom but, as Section 5 shows, does not prevent a magnitude penalty from biasing the interior toward the boundary-induced origin. Proposition 3 and Lemma 1 are proved in Section B.

5

Graph-Sobolev Regularization

We show that ridge (standard-form Tikhonov) regularization inverts the recovered ordering (Section 5.1), introduce the gauge-invariant Dirichlet-energy penalty (Section 5.2), and establish three properties: gauge invariance (Section 5.3), an SPD reduced system (Section 5.4), and exact range preservation on chains (Section 5.5). Section 5.7 reports the headline parameter-insensitivity: the estimate is stable across four orders of magnitude in λ1 , ridge only at λ1 → 0. Proofs are in Section B. 5.1

The ridge inversion and the gauge mismatch

The standard estimator augments Eq. (2) with a magnitude penalty, 2

2

ϕ̂Tik = arg min 12 ∥Lϕ − b∥2 + λ1 ∥ϕ∥2 ,

(3)

ϕ

subject to the Dirichlet boundary. The penalty pulls every interior value toward 0. But on a Dirichlet problem 0 is not distinguished: it is merely where the boundary placed the abandon sink. The estimator therefore has a bias toward the abandon boundary set by the boundary labelling, not the data, and Proposition 4 makes its effect exact: the interior range shrinks like 1/λ1 , so the ordering flattens and then reverses. Proposition 4 (Magnitude shrinkage collapses the range). Let ϕ̂Tik (λ1 ) solve Eq. (3) on an interior chain of length L with the sink pinned at 1 and unit divergence injected at the free source. Then the interior dynamic range ∆ϕ̂Tik = maxI ϕ̂ − minI ϕ̂ satisfies ∆ϕ̂Tik (λ1 ) ≤ C/λ1 → 0 as λ1 → ∞, with a constant C independent of the chain length L: the magnitude penalty extracts no benefit from a longer chain. 4

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

Proposition 4 does not claim the rank correlation tends to −1; empirically it settles to a negative plateau near −0.42 (Figure 1), because the collapsed order is dominated by residual divergence structure anticorrelated with the planted field. The damage is decisive: ridge inverts the ordering for every λ1 > 0, linear correlation crossing to zero (Table 1). 5.2

The graph-Sobolev penalty

Definition 1 (Graph-Sobolev objective). Let LG be the symmetric, conductance-weighted graph Laplacian of the undirected support of Gδ , so that X ϕ⊤ LG ϕ = wuv (ϕv − ϕu )2 . (4) (u,v)∈E

The graph-Sobolev estimator is 2

ϕ̂Sob = arg min 12 ∥Lϕ − b∥2 + λ1 ϕ⊤ LG ϕ,

(5)

ϕ

subject to the Dirichlet boundary. We use the weighted energy: each edge contributes wuv (ϕv − ϕu )2 with the flux conductance of Eq. (1) (the penalty in the released code; every number uses it). The unweighted incidence form LG = G⊤ G is the case wuv ≡ 1. 5.3

Gauge invariance

A regularizer R is gauge-invariant if R(ϕ + c1) = R(ϕ) for all c ∈ R. Proposition 5 (Gauge invariance). The Dirichlet energy ϕ⊤ LG ϕ is gauge-invariant, whereas the 2 magnitude penalty ∥ϕ∥2 is not. The conceptual heart: since LG 1 = 0 the Dirichlet energy penalizes only differences and is indifferent 2 to the origin, whereas ∥ϕ∥2 imposes a preferred mean-zero gauge that has no meaning in the Dirichlet problem and fights the boundary conditions. 5.4

Well-posedness

Theorem 1 (SPD reduced system). Suppose every interior node has a directed path to the boundary in Gδ . Then for every λ1 > 0 the reduced graph-Sobolev operator
M = L⊤ L II + λ1 (LG )II (6) is symmetric positive definite, so Eq. (5) has a unique interior solution. The added PSD term also improves the conditioning of the unregularized operator without disturbing the gauge. Under this hypothesis M is SPD and a Cholesky factorization solves Eq. (5) directly. In practice, however, many interior nodes have no path to a sink, so the hypothesis fails and M is only positive semidefinite; we therefore solve by a symmetric eigensolve that inverts only the nonzero spectrum, giving the minimum-norm solution. This keeps the gauge mode (in the range of M ) exact and leaves undetermined nodes at the smooth default, excluded from scoring (Section 5.7); a Cholesky solve with a diagonal floor would instead amplify roundoff along the near-null directions. 5.5

Range preservation on chains

Theorem 2 (Range preservation, exact). On the length-L constant-conductance chain of Proposition 4, with the trace-normalized Dirichlet-energy penalty (Section 5.6), the graph-Sobolev estimator has interior dynamic range exactly 2L + 1 1 2 = , κ(L) = = O(1/L). (7) ∆ϕ̂Sob (λ1 ) = 2L + 1 + 2λ1 1 + λ1 κ(L) 2L + 1 Hence for any fixed λ1 the retained range tends to the full harmonic range ∆ϕtrue = 1 as L → ∞. Contrasted with Proposition 4, the Sobolev range depends on λ1 only through λ1 /(2L + 1), whereas ridge collapses like 1/λ1 independently of L: a longer chain helps the Dirichlet-energy estimate and never helps ridge. On a length-30 chain, Eq. (7) gives ∆ϕ̂Sob = 61/63 = 0.968 at λ1 = 1 and 61/81 = 0.753 at λ1 = 10, versus 0.536 and 0.091 for ridge (Figure 4, Section B). 5

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

Table 1: Recovery and top-5 ranking power versus λ1 (determined interior, n = 95; HHD extraction). The gauge-invariant penalty is insensitive to λ1 (rank correlation constant at +0.81 and NDCG@5 in [0.97, 1.0] across four orders of magnitude), while ridge inverts the ordering for every λ1 > 0 (Spearman to ≈−0.42, NDCG@5 from 0.97 to 0.76). Metrics vs. the planted ϕtrue (graded relevance for NDCG); Spearman uses estimates rounded to tolerance so the saturated near-zero cluster does not break ties by roundoff. graph-Sobolev

Tikhonov

λ1

Spearman

Pearson

NDCG@5

Spearman

Pearson

NDCG@5

0 10−3 10−2 10−1 1 10

+0.810 +0.807 +0.807 +0.807 +0.807 +0.807

+0.758 +0.760 +0.767 +0.783 +0.831 +0.849

0.966 0.966 0.982 0.982 0.999 0.982

+0.810 −0.450 −0.424 −0.418 −0.422 −0.551

+0.758 −0.055 −0.005 +0.001 +0.001 +0.001

0.966 0.756 0.756 0.756 0.756 0.756

graph-Sobolev

Tikhonov

Spearman ( , true)

Pearson r( , true)

0.75 0.50 0.25 0.00 0.25 0.50

0.75 0.50 0.25 0.00 0.25 0.50 0

10 3 10 2 10 1 100 regularization strength 1

0

101

10 3 10 2 10 1 100 regularization strength 1

101

Figure 1: Recovery of the planted potential versus λ1 . Graph-Sobolev (blue) preserves both rank and linear correlation across four orders of magnitude; Tikhonov (red) inverts the rank correlation to ≈ − 0.42 and collapses the linear correlation to zero. Determined interior, n = 95.

5.6

Trace normalization and cost

To make λ1 comparable across graphs, we normalize the penalty block by its trace, λ1 (LG )II 7→ λ1 (LG )II / tr[(LG )II ]; all sweeps use this. The dense solve is O(|I|3 ): Cholesky would suffice under Theorem 1 (M SPD), but real extractions violate that hypothesis, so we use the symmetric eigensolve of Section 5.4 throughout to return the minimum-norm solution. It is negligible for the tens-to-hundreds of interior states here; a conjugate-gradient solve at O(κ(M ) nnz) is available when |I| is large. 5.7

Parameter-insensitivity: preservation versus collapse

We compare the two penalties on the synthetic instrument of Section 7 (planted harmonic ϕtrue , |V | = 277, five sinks), scoring on the determined interior: the n = 95 well-visited states (≥ 100 visits) with a directed path to the boundary (nodes without one are undetermined by the data, Theorem 1). Rank correlation is computed on estimates rounded to numerical tolerance so the saturated near-zero cluster (Lemma 1) ties deterministically rather than by roundoff. Figure 1 and Table 1 report the headline—robustness to λ1 , not a single best setting. At λ1 = 0 both estimators recover the planted ordering (ρs = +0.81, r = +0.76); as λ1 grows, graph-Sobolev holds this unchanged (rank correlation constant at +0.807 across λ1 ∈ [10−3 , 10], r ∈ [+0.76, +0.85]), while ridge inverts the instant regularization is applied, dropping to a negative plateau near −0.42 (linear correlation crossing zero). The ≈ + 1.23 gap is itself flat in λ1 —no ridge setting recovers the lost structure (the pointwise collapse is Figure 3, Section C). The same holds for top-k power: NDCG@5 stays in [0.97, 1.0] across the whole λ1 range under Sobolev but drops to 0.76 for ridge (Table 1), so parameter-insensitivity shows in top-of-ranking quality, not only correlation. 6

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

6

Boundary Identification from Flow Geometry

When the absorbing boundary is unknown, ranking states by the recovered potential fails on multi-sink graphs: with several sinks pinned at the same value, the interior potential is rank-deficient and sinks do not separate from high-potential interior states. The Poisson residual ru = |[Lϕ]u − bu | succeeds instead. Theorem 3 (Residual concentration at absorbing nodes). Let u be an absorbing sink with no outgoing DAG edges. Then [Lϕ]u = 0 exactly, so ru = |bu | equals the empirical inflow at u. If every interior node satisfies the Poisson equation to tolerance η and minu∈B |bu | > η, then every sink ranks strictly above every interior node under r. A sink is where mass accumulates without a modeled outflow: the forward operator predicts zero net flux but the data shows large inflow, and that mismatch is the residual; interior nodes, where the solve balances flux, have small residual. On the synthetic instrument (|V | = 277, five planted sinks) the residual ranks all five true sinks in the top seven states (worst rank 7), while potential ranking scatters them to worst rank 108; the separation is invariant across abandonment rates {0.05, 0.20, 0.50}, making the residual a regime-invariant boundary detector. Theorem 3 is proved in Section B.

7

Synthetic Validation

We validate on a controlled instrument: recovery can only be scored against a known potential, which no observational corpus provides. The instrument plants a ground-truth potential, samples flow from it, and hands the algorithm only the flow; every number comes from the released generator at one fixed configuration. The instrument. The generator builds a multi-branch conversion funnel: a source feeds shared upper states; each of K = 5 branches is a chain terminating in a conversion sink; one abandon sink absorbs drop-off at a depth-decaying rate. The ground-truth potential is the harmonic extension under {conversion = 1, abandon = 0}, i.e. ϕtrue (u) = Pr[absorbed at a conversion sink before abandon | start at u], exactly the object the Poisson solve targets. The default point uses 20 shared states, 50 interior states per branch, abandonment 0.05, and 50,000 sessions, giving |V | = 277 and terminalsink entropy H(π) ≈ 2.27 bits; sessions are reproducible bit-for-bit at a given seed. Structural certificates and recovery. The extracted operator is nilpotent (∥Am ∥F = 0 at the predicted m = 7), confirming Proposition 2 (no residual cycle). The Hodge gradient component ϕ0 alone does not recover the planted direction (conductance-weighted cosine −0.06): an honest negative showing the gradient projection is a structural device, not an estimator. Recovery is the regularized solve’s job (Section 5.7): graph-Sobolev holds rank correlation +0.81 while ridge inverts to ≈ − 0.42, and preserves 97% of the chain range against ridge’s 54% (Figure 4), independent of the Hodge step (Section 7.1). 7.1

Does the Hodge projection matter? An extraction ablation

We ask whether any acyclic support—in particular a cheap topological sort— serves as well as the Hodge projection, replacing it with two dominance-thresholded topological sorts, the rest of the pipeline fixed. The P orders come from data alone: visit-position (mean fractional session position) and net-inflow ( v (Fvu − Fuv )); an edge is kept when net-forward, dominant, and order-consistent (acyclic by construction). Table 2 reports recovery under each extraction. Two conclusions follow. First, the regularizer contrast, the paper’s central claim, is invariant to the extraction: graph-Sobolev recovers a strong positive ordering under all three supports while ridge inverts under all three (ρs = −0.42, −0.32, −0.56; margin +1.23 to +1.52). Second, Hodge is not the best extraction for recovery: both topological sorts give higher Sobolev rank correlation (+0.96, +0.99) than Hodge (+0.81). We treat the topological sort as the default, keeping Hodge for its generality (only the flow matrix) and the nilpotency certificate, not accuracy. Either way, the contribution is isolated in the regularizer, not a Hodge artifact. 7

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

Table 2: Extraction ablation (λ1 = 1, determined interior: n = 95 for Helmholtz–Hodge, n = 150 for the topological sorts, which reach the boundary from every well-visited state). The regularizer contrast holds under every acyclic support: graph-Sobolev recovers a strong positive ordering while ridge inverts. Hodge is not required for recovery; all supports are exactly acyclic. Spearman vs. ϕtrue Extraction

edges

graph-Sobolev

Tikhonov

margin

Helmholtz–Hodge (Section 3) Topological sort (visit-position) Topological sort (net-inflow)

185 664 375

+0.807 +0.995 +0.961

−0.422 −0.324 −0.556

+1.23 +1.32 +1.52

Table 3: Real corpora, reduced to event-type state spaces. Conversion is the fraction of sessions terminating at a conversion sink.

7.2

Dataset

Sessions

States

Conversion

Conversion sink

RetailRocket Trivago OTTO

41,184 640,728 186,271

4 11 105

33.0% 92.6% 27.3%

transaction clickout item orders

Regime dependence and scope

The magnitude of the recoverable correlation is a regime property, not the regularizer’s: at low abandonment the planted field is compressed and the unregularized solve recovers ρs ≈ +0.81, while as abandonment rises the well-visited interior shrinks and recovery degrades (to ρs ≈ +0.23 at abandonment 0.5). The regularizer’s role is invariant throughout: graph-Sobolev preserves whatever ordering the unregularized solve attains and ridge degrades it; what is regime-dependent is whether there is signal to preserve. We confirm this is not an artifact of the single configuration of Table 1: across twelve instrument variants—five seeds plus single-axis perturbations of branch count, chain depth, abandonment, and sink entropy—the Sobolev-minus-Tikhonov rank margin at λ1 = 1 is positive in all twelve under both extractions (Table 7, Section C). On the topological-sort support the ordering is configurationinvariant (ρs = +0.99 ± 0.00 vs. ridge −0.34 ± 0.06); on the Hodge support the Sobolev correlation tracks the regime (down to +0.23 at abandonment 0.5) yet stays positive, with margin never below +0.45. 7.3

Scaling to 104 nodes

Because the reduced system is symmetric PSD (Theorem 1), the solve admits sparse conjugate gradients (CG), which converge to the same minimum-norm estimate as the dense path. On a planted funnel with |V | = 10,202 (roughly 104 nodes), graph-Sobolev recovers ρs = +0.94 while ridge inverts to −0.18, and CG returns in 0.08 s where the dense solve is killed for memory. Table 4 (Section C) times CG against the dense eigensolve: CG matches it to 10−9 , scales near-linearly, and handles 4 × 104 nodes in 0.39 s (dense exhausts memory beyond ∼4,000), using the topological-sort extraction of Section 7.1.

8

Real-Data Validation

No clickstream corpus carries a ground-truth potential, so we validate claims that need no planted field: the recovered potential is interpretable (Section 8.1); the regularizer contrast holds internally (Section 8.2); and the residual concentrates at the known boundary while ϕ is stable under resampling (Section 8.3). We use three public corpora, each reduced to an event-type state space (Section 3) and summarized in Table 3: RetailRocket (a tiny funnel), Trivago (a medium action graph), and OTTO (a large multi-sink graph). 8

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

8.1

Recovered potentials are interpretable

The graph-Sobolev potential (λ1 = 1) reproduces the known funnel ordering (Figure 5, Section C). On RetailRocket, view (−0.60) sits below the abandon origin and addtocart (+0.32) between the boundaries; on Trivago, clickout is highest and searches lowest, with one flagged anomaly (interaction item image, −1.52). On OTTO (the only genuinely multi-sink graph) the solve separates carts (mean +0.26, straddling the cart sink) from clicks (mean −0.01), recovering clicks < carts < orders from flow alone. 8.2

The regularizer contrast holds without ground truth

With no truth, we measure how each penalty at λ1 = 1 perturbs the unregularized solve along two axes: interior ordering (Spearman vs. the base) and dynamic range (fraction retained). Table 5 and Figure 2 reproduce the synthetic finding cleanly: graph-Sobolev retains 28–41% of the range across all three corpora, while ridge retains 21%, 5%, and, on the 105-state OTTO graph, just 0.2%. The contrast sharpens with graph size; on OTTO ridge retains essentially none (rank agreement 0.42 vs. graph-Sobolev’s 0.66), as Proposition 4 predicts. 8.3

Residual concentration and stability

The Poisson residual ranks the known conversion sink in the top one to three of up to 105 states (RetailRocket 3/4, Trivago 1/11, OTTO 2/105), confirming Theorem 3; being single-sink, these corpora do not exercise its multi-sink advantage over potential ranking. Bootstrap resampling (20 resamples) gives small coefficients of variation on well-visited states (median 0.010, 0.021, 0.127); ϕ is stable wherever the divergence is well-sampled. 8.4

The recovered potential is a usable node feature

Does the potential carry downstream signal? We predict OTTO session conversion with a logistic regression on four per-session order statistics —mean, max, min, last—of the interior states’ potentials; the task is leakage-guarded, as the label is the terminal sink, the features exclude every sink, and the potential is fit on the training sessions only (Section C). The graph-Sobolev potential adds +3.1 ROC-AUC points over raw session statistics (0.828 → 0.859; Table 6) while ridge adds nothing (0.829); standalone, the 4-D Sobolev feature scores 0.836 against ridge’s 0.717. The gauge-invariant penalty yields a usable (if compressed) node feature where the collapsed ridge potential does not (Table 6).

9

Conclusion

We showed that the standard regularizer for directed Poisson inverse problems fails specifically: the ridge (identity-seminorm) penalty imposes a meaningless origin and, as its strength grows, collapses the dynamic range and reverses the ordering, so the only safe setting is no regularization. The gauge-invariant Dirichlet energy removes the hazard, with one defining consequence, parameterinsensitivity: the recovered potential is stable across four orders of magnitude in λ1 , whereas ridge inverts the ordering for every λ1 > 0. We proved gauge invariance (Proposition 5), an SPD reduced system (Theorem 1), and exact range preservation on chains (Theorem 2); gave a Poisson-residual boundary diagnostic; and positioned the contribution against general-form Tikhonov and graph smoothness priors. The penalty is classical, but its necessity and parameter-insensitivity here are not. The guarantee is preservation, not signal: whatever ordering the data supports, the gauge-invariant penalty keeps it across λ1 and ridge does not. Future work. Open directions: characterizing when the divergence carries enough of the harmonic field for recovery (a theorem for the regime dependence of Section 7.2); nonlinear constitutive laws; edge-preserving gauge-invariant penalties (total variation, the p-Laplacian); and which acyclic supports best aid recovery (Section 7.1). All experiments, figures, and tables are reproduced by the released code from a single fixed-seed command. 9

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

References Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6):1373–1396, 2003. 2 Cristian Bodnar, Francesco Di Giovanni, Benjamin P. Chamberlain, Pietro Liò, and Michael M. Bronstein. Neural sheaf diffusion: A topological perspective on heterophily and oversmoothing in GNNs. In Advances in Neural Information Processing Systems (NeurIPS), volume 35, pages 18527–18541, 2022. 3 Peter G. Doyle and J. Laurie Snell. Random Walks and Electric Networks. Carus Mathematical Monographs. Mathematical Association of America, 1984. 2 Leo Grady. Random walks for image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(11):1768–1783, 2006. 2 Per Christian Hansen. Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadelphia, 1998. 2 Yixuan He, Michael Perlmutter, Gesine Reinert, and Mihai Cucuringu. MSGNN: A spectral graph neural network based on a novel magnetic signed Laplacian. In Learning on Graphs Conference (LoG), volume 198 of PMLR, 2022. 3 Junteng Jia, Michael T. Schaub, Santiago Segarra, and Austin R. Benson. Graph-based semisupervised and active learning for edge flows. In Proceedings of the 25th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD), pages 761–771, 2019. 2, 3 Xiaoye Jiang, Lek-Heng Lim, Yuan Yao, and Yinyu Ye. Statistical ranking and combinatorial Hodge theory. Mathematical Programming, 127(1):203–244, 2011. 3 Lek-Heng Lim. Hodge Laplacians on graphs. SIAM Review, 62(3):685–715, 2020. 3 Antonio G. Marques, Santiago Segarra, and Gonzalo Mateos. Signal processing on directed graphs: The role of edge directionality when processing and learning from network data. IEEE Signal Processing Magazine, 37(6):99–116, 2020. 3 Emanuele Rossi, Bertrand Charpentier, Francesco Di Giovanni, Fabrizio Frasca, Stephan Günnemann, and Michael M. Bronstein. Edge directionality improves learning on heterophilic graphs. In Learning on Graphs Conference (LoG), volume 231 of PMLR, 2023. 3 Leonid I. Rudin, Stanley Osher, and Emad Fatemi. Nonlinear total variation based noise removal algorithms. Physica D: Nonlinear Phenomena, 60(1–4):259–268, 1992. 2, 3 Michael T. Schaub and Santiago Segarra. Flow smoothing and denoising: Graph signal processing in the edge-space. In IEEE Global Conference on Signal and Information Processing (GlobalSIP), pages 735–739, 2018. 2, 3 Michael T. Schaub, Austin R. Benson, Paul Horn, Gabor Lippner, and Ali Jadbabaie. Random walks on simplicial complexes and the normalized Hodge 1-Laplacian. SIAM Review, 62(2):353–391, 2020. 3 David I. Shuman, Sunil K. Narang, Pascal Frossard, Antonio Ortega, and Pierre Vandergheynst. The emerging field of signal processing on graphs. IEEE Signal Processing Magazine, 30(3):83–98, 2013. 2 Andrey N. Tikhonov and Vasiliy Y. Arsenin. Solutions of Ill-Posed Problems. Winston & Sons, Washington, D.C., 1977. 2 Yu-Xiang Wang, James Sharpnack, Alexander J. Smola, and Ryan J. Tibshirani. Trend filtering on graphs. Journal of Machine Learning Research, 17(105):1–41, 2016. 2, 3 Xitong Zhang, Yixuan He, Nathan Brugnone, Michael Perlmutter, and Matthew Hirn. MagNet: A neural network for directed graphs. Advances in Neural Information Processing Systems (NeurIPS), 34:27003–27015, 2021. 3 Lingxiao Zhao and Leman Akoglu. PairNorm: Tackling oversmoothing in GNNs. In International Conference on Learning Representations (ICLR), 2020. 3, 14 Dengyong Zhou, Olivier Bousquet, Thomas N. Lal, Jason Weston, and Bernhard Schölkopf. Learning with local and global consistency. In Advances in Neural Information Processing Systems (NeurIPS), volume 16, pages 321–328, 2004. 2 10

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

Kaixiong Zhou, Xiao Huang, Daochen Zha, Rui Chen, Li Li, Soo-Hyun Choi, and Xia Hu. Dirichlet energy constrained learning for deep graph neural networks. In Advances in Neural Information Processing Systems (NeurIPS), 2021. 3 Xiaojin Zhu, Zoubin Ghahramani, and John Lafferty. Semi-supervised learning using Gaussian fields and harmonic functions. In Proceedings of the 20th International Conference on Machine Learning (ICML), pages 912–919, 2003. 2

A

Extraction Mechanics

This appendix gives the edge-retention rule deferred from Section 3, for both orientations. Write the oriented coordinate as ψ: for the topological sort ψ is the mean visit position (or net inflow); for the Helmholtz–Hodge projection ψ = ϕ0 , the gradient potential solving G⊤ G ϕ0 = G⊤ ω. With the gradient-dominance ratio |ψ(v) − ψ(u)| δ(u, v) = , ωuv = Fuv − Fvu , (8) |ωuv | + ϵ the retained edge set is  Eδ = (u, v) : ψ(v) > ψ(u) and δ(u, v) ≥ τ , (9) followed by top-k pruning per source (retain each node’s k heaviest outgoing edges). The threshold τ trades support density against strictness: a larger τ keeps only edges whose oriented drop dominates the raw flow, and on small dense graphs (for example the eleven-state event graph of Section 8) a lower τ is needed to avoid disconnecting interior states. All reported runs use ρ = 2, k = 10, and τ as stated per experiment.

B

Proofs

The chain instrument. Both proofs use the chain implemented in the released code. The nodes are 0, 1, . . . , L + 1 with unit-conductance edges i → i + 1 for i = 0, . . . , L. The sink L + 1 is the only Dirichlet node, pinned at ϕL+1 = 1; unit divergence is injected at the source and absorbed at the sink, b0 = −1,

bL+1 = +1,

bi = 0 (1 ≤ i ≤ L).

The interior unknown is x = (ϕ0 , . . . , ϕL ). The directed Laplacian acts as (Lϕ)i = ϕi − ϕi+1 for i = 0, . . . , L (and (Lϕ)L+1 = 0 since the sink has no out-edge). Writing the interior block LII and eliminating the pinned sink gives the effective right-hand side beff = bI − LIB ϕB , which has a −1 in coordinate 0 and a +1 in coordinate L and is zero elsewhere. It is convenient to use the edge gaps PL di = ϕi − ϕi+1 , i = 0, . . . , L, so ϕi = 1 + j=i dj , since ϕL+1 = 1. A short computation gives (LII x − beff )0 = d0 + 1, ( · )i = di for 1 ≤ i ≤ L, so L X eff 2 2 LII x − b 2 = (d0 + 1) + d2i , i=1

⊤

ϕ LG ϕ =

L X

d2i .

(10)

i=0

The interior dynamic range is ∆ϕ̂ = max0≤i≤L ϕi − min0≤i≤L ϕi . B.1

Proof of Proposition 4 2

2

Ridge minimizes 12 LII x − beff 2 + λ1 ∥x∥2 (up to the positive data-normalization constant, which ⊤ eff only rescales λ1 ), with normal equations (A + λ1 I)x = c, where A = L⊤ II LII ⪰ 0 and c = LII b . Because A ⪰ 0, the matrix (A + λ1 I)−1 has spectral norm at most 1/λ1 , so ∥c∥2 2 ∥c∥2 , hence ∆ϕ̂Tik ≤ 2 ∥x∥∞ ≤ 2 ∥x∥2 ≤ . λ1 λ1 It remains to bound ∥c∥2 independently of L. The matrix LII is upper bidiagonal with 1 on the diagonal and −1 on the superdiagonal (and a lone 1 in the last row), so column j of LII eff eff eff has entries (LII )jj = 1 and (LII )j−1,j = −1. Thus cj = (L⊤ II b )j√= bj − bj−1 , and with eff ⊤ b = (−1, 0, . . . , 0, 1) this gives c = (−1, 1, 0, . . . , 0, 1) , so ∥c∥2 = 3 for every L. Therefore √ ∆ϕ̂Tik (λ1 ) ≤ 2 3/λ1 → 0, with a constant independent of L. ∥x∥2 = (A + λ1 I)−1 c 2 ≤

11

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

B.2

Proof of Theorem 2

With the trace normalization of Section 5.6, the Sobolev objective is f (x) =

L L i X 1 h λ1 X 2 (d0 + 1)2 + d , d2i + 2 bsc 2 τL i=0 i i=1 2

using Eq. (10), where bsc = beff 2 = 2 is the data-normalization constant and τL = tr[(LG )II ] = 1 + 2(L − 1) + 2 = 2L + 1 is the penalty trace (node 0 has degree 1, nodes 1, . . . , L have degree 2). The crucial point is that f is separable in the gaps d0 , . . . , dL : each di appears in exactly one data term and one penalty term. Minimizing term by term, d⋆i = 0 (1 ≤ i ≤ L),

d⋆0 = arg min d

(d + 1)2 λ1 2 −1 −(2L + 1) + d = = , 2bsc 2τL 1 + λ1 bsc /τL 2L + 1 + 2λ1

where we used bsc = 2. Since d⋆1 = · · · = d⋆L = 0, the recovered potential is constant on {1, . . . , L + 1} and drops by |d⋆0 | at the source, so ∆ϕ̂Sob (λ1 ) = |d⋆0 | =

2L + 1 1 = , 2L + 1 + 2λ1 1 + λ1 κ(L)

κ(L) =

2 . 2L + 1

1 The bound 1+λ11κ(L) ≥ 1 − λ1 κ(L) follows from 1+t ≥ 1 − t for t ≥ 0, and ∆ϕtrue = 1 is the

λ1 = 0 value. As L → ∞, κ(L) → 0 and ∆ϕ̂Sob → 1. Numerical confirmation. The released solver reproduces ∆ϕ̂Sob = (2L + 1)/(2L + 1 + 2λ1 ) to four decimals at every tested (L, λ1 ) (for example 61/63 = 0.968 at (L, λ1 ) = (30, 1) and 61/81 = 0.753 at (30, 10)), and confirms that ∆ϕ̂Tik is independent of L (identical to four decimals for L = 30 and L = 100 at each λ1 ), as Proposition 4 predicts. B.3

Remark on the empirical Tikhonov plateau

Proposition 4 gives ∆ϕ̂Tik → 0 but makes no claim about the limiting rank correlation. On a graph with nonzero interior divergence the collapsed estimate is not identically constant: the O(1/λ1 ) leading term is ∝ c = L⊤ beff , whose sign pattern reflects local divergence rather than global position between boundaries. The rank correlation with ϕtrue therefore tends to a regime-dependent negative plateau (empirically ≈ − 0.42 in Table 1), not to −1: the recovered order is anticorrelated with the planted field but not its exact reverse. The linear correlation likewise crosses to zero as the range collapses (Table 1). B.4

Proofs of the structural results

Proof of Proposition 1 (acyclicity). By Eq. (9), every retained edge (u, v) has ψ(v) > ψ(u) for the oriented coordinate ψ. A directed cycle v1 → · · · → vk → v1 would force ψ(v1 ) < · · · < ψ(vk ) < ψ(v1 ), a contradiction; hence ψ strictly increases along every directed path and is a topological order. Proof of Proposition 2 (nilpotency). (Am )uv sums over directed walks of length m from u to v. On a DAG every walk is a path, of length at most L; hence for m > L no such walk exists and Am = 0. Proof of Proposition (minimum dissipation). Introduce a multiplier ϕ for the divergence constraint P 3 −1 2 and form L = 12 Wij qij + ϕ⊤ (div(q) − b). Stationarity in qij gives Wij−1 qij = ϕi − ϕj up to orientation, i.e. qij = Wij (ϕj − ϕi ); substituting into div(q) = b yields Lϕ = b. Proof of Lemma 1 (chain saturation). Zero divergence at vi means inflow equals outflow, Wvi−1 vi (ϕvi−1 − ϕvi ) = Wvi vi+1 (ϕvi − ϕvi+1 ), which telescopes; with a single equal-weight inand out-edge this forces ϕvi = ϕvi+1 for all i, and pinning the terminal value propagates ϕvi = ϕ(s) backward. 12

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

Proof of Proposition 5 (gauge invariance). Since LG 1 = 0, (ϕ + c1)⊤ LG (ϕ + c1) = ϕ⊤ LG ϕ + 2 2 2c 1⊤ LG ϕ + c2 1⊤ LG 1 = ϕ⊤ LG ϕ. By contrast ∥ϕ + c1∥2 = ∥ϕ∥2 + 2c 1⊤ ϕ + c2 |V | depends on ⋆ c, with a unique minimizing gauge c = −ϕ̄. Proof of Theorem 1 (SPD reduced system). M is symmetric. For x ̸= 0 on the interior, LII has trivial null space: LII x = 0 makes x constant on each interior component, and the path-to-boundary hypothesis forces that constant to match a pinned boundary value, so x = 0; hence x⊤ (L⊤ L)II x = 2 ∥LII x∥2 > 0. The Dirichlet block (LG )II is PSD, and a positive-definite plus a PSD matrix is positive definite, so x⊤ M x > 0. Proof P of Theorem 3 (residual concentration). For an absorbing sink u with no out-edges, [Lϕ]u = j:(u,j)∈E Wuj (ϕu − ϕj ) has no terms, so it is 0 and ru = |bu |. For an interior node ru = |[Lϕ]u − bu | ≤ η by hypothesis, and minB |bu | > η then orders all sinks above all interior nodes.

C

Additional Figures and Tables

Table 4: Solver scaling: sparse conjugate gradients (CG) versus the dense eigensolve for the graphSobolev solve (λ1 = 1) on layered DAGs of growing interior size. CG matches the dense solve to 10−9 where both run, scales near-linearly in edges, and handles 4 × 104 nodes in under half a second; the dense solve exhausts memory beyond ∼4,000 interior nodes. This backs the CG path of Section 7.3 (Theorem 1). Times are single-core wall-clock and reproduced by the released run_scaling script. interior nodes

edges

CG (s)

eigensolve (s)

max|∆|

1,000 2,000 4,000 8,000 16,000 40,000

1,980 3,960 7,920 15,840 31,840 79,800

0.010 0.015 0.026 0.054 0.121 0.392

0.16 1.19 12.89 out of memory out of memory out of memory

2.7 × 10−9 4.0 × 10−9 3.0 × 10−9 n/a n/a n/a

Table 5: Regularizer contrast on real data at λ1 = 1, versus the unregularized solve (exact values for Figure 2). “Range ret.” is interior dynamic range as a fraction of the base; “Rank agree.” is the Spearman correlation of the interior ordering with the base. graph-Sobolev

Tikhonov

Dataset

states

Range ret.

Rank agree.

Range ret.

Rank agree.

RetailRocket Trivago OTTO

4 11 105

0.407 0.276 0.279

1.000 0.883 0.662

0.208 0.053 0.002

1.000 0.833 0.419

Downstream protocol (Table 6). Each session is labelled by its terminal sink (converted iff it ends at orders) and featurised only from the interior (non-sink) states it visits, so the pinned label cannot leak into the features. The potential features are the four order statistics—mean, max, min, and last-visited ϕ̂—over those states; the raw baseline is [#visits, #distinct states], and “raw+” rows concatenate the two. The classifier is an L2 -regularised logistic regression (scikit-learn defaults, C=1); we hold out 30% of sessions and average over three random splits, and—to avoid leakage across the split—fit the potential on the training sessions only, then apply it to both. Because ϕ̂ is a function of the state, the potential features are a 1-D-per-state compression of node identity, and therefore sit below the bag-of-states ceiling (0.899) by construction: they cannot exceed the full 105-dimensional identity. The finding is not that they do, but that the gauge-invariant compression recovers about a third of the raw-to-ceiling AUC gap in four features while the ridge compression recovers essentially none; a learned encoder (e.g. GraphSAGE) on ϕ̂ could close the remainder. Reproduced by the released conversion_prediction experiment. 13

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

Table 6: Downstream utility on OTTO: session-level conversion prediction (Section 8.4), held-out ROC-AUC (mean±sd over three splits; 27% conversion base rate, 186,271 sessions). Features use only the interior (non-sink) states a session visits; the label is the terminal sink, so the task is leakage-free. The graph-Sobolev potential is a usable node feature—adding +3.1 points over raw statistics and beating the ridge potential by +11.8 points standalone—whereas the collapsed ridge potential adds nothing. A trained directed GNN (run_gnn, mean/in/out message passing, node embeddings pooled over the same visited states) is stronger (0.861) but oversmooths with depth, its AUC falling to 0.807 at 32 layers as node-embedding Dirichlet energy collapses; neutralizing the constant/gauge mode each layer holds it flat (0.854)—the gauge principle of Section 2 applied to the network, recovering PairNorm (Zhao and Akoglu, 2020). Bag-of-states (full 105-dim visit counts) is the state-identity ceiling. Architecture and gauge-centering in Section C; reproduced by the released conversion_prediction and run_gnn experiments. Feature set

test ROC-AUC

Raw session statistics (length, #distinct) + graph-Sobolev potential (4-D) + Tikhonov potential (4-D) graph-Sobolev potential alone (4-D) Tikhonov potential alone (4-D)

0.828 ± 0.001 0.859 ± 0.000 0.829 ± 0.001 0.836 ± 0.000 0.717 ± 0.001

Directed GNN, 2 layers (trained) Directed GNN, 32 layers (oversmoothed) Directed GNN, 32 layers (gauge-centered) Bag-of-states ceiling (105-D)

0.861 0.807 0.854 0.899 ± 0.001

Directed-GNN baseline and gauge-centering (Table 6). The GNN rows use a directed graph convolutional network on the extracted DAG Gδ (topological-sort support, N = 105 nodes). With learnable node embeddings H 0 ∈ RN ×d (d = 32), each layer aggregates from in- and out-neighbours separately,   H l+1 = ReLU Âout H l Wol + Âin H l Wil , (11) where Âout is the row-normalized adjacency of Gδ and Âin that of its transpose (self-loops enter through the pure propagation of Section 2). A session is scored by mean-pooling the final embeddings over the interior states it visits—the same pooling and leakage guard as the potential features (above)— followed by a logistic layer; we train end-to-end with Adam (learning rate 10−2 , weight decay 10−4 , 80 epochs, binary cross-entropy) on the same 70/30 session split. Oversmoothing drives H l toward the constant (gauge) mode, the smallest eigenvector of the symmetric normalized Laplacian Lsym , which is u0 ∝ d1/2 (node degrees d). Gauge-centering removes this component after every layer,
l H l ← H l − u0 u⊤ ∥u0 ∥2 = 1, (12) 0H , i.e. it subtracts the degree-weighted mean embedding—the per-layer analogue of LG 1 = 0 (for a regular graph u0 ∝ 1 and this is the ordinary mean). The “gauge+scale” variant additionally rescales H l to fixed Frobenius norm, recovering PairNorm (Zhao and Akoglu, 2020). Oversmoothing is measured by the node Dirichlet energy E(H) = tr(H ⊤ Lsym H)/∥H∥2F , which vanishes as representations collapse. Across depths {2, 4, 8, 16, 32} the vanilla GCN’s test AUC falls 0.86 → 0.81 as E collapses (to 0.52 at 16 layers), while gauge-centering holds AUC flat (0.85 at 32 layers). Reproduced by the released run_gnn experiment.

D

Uncertainty Quantification

The divergence b is an empirical estimate, and its sampling covariance propagates to ϕ in closed form. eff On the reduced interior system ϕI = L−1 II bI (at λ1 = 0 for clarity), the delta method gives −⊤ d Cov(ϕI ) = L−1 II Cov(bI ) LII ,

(13)

−1 ⊤ d with Cov(b) estimated from the multinomial transition counts. For λ1 > 0 replace L−1 LII II by M with M as in Eq. (6).

14

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

0.5 0.4

Interior dynamic range retained (fraction of unregularized range) 0.407

0.3 0.2

1.0

0.279

0.6

0.276 0.208

0.1 0.0

Rank agreement with the unregularized solve (Spearman)

graph-Sobolev Tikhonov

0.4 0.2

0.053

RetailRocket (4 states)

0.8

Trivago (11 states)

0.002

OTTO (105 states)

0.0

RetailRocket (4 states)

Trivago (11 states)

OTTO (105 states)

Figure 2: Regularizer contrast on real data (λ1 = 1) versus the unregularized solve. Left: interior dynamic range retained. Right: rank agreement with the base solve. Ridge’s collapse deepens as the graph grows, reaching near-total loss on OTTO; graph-Sobolev preserves both throughout. Exact values in Table 5. Table 7: Robustness of the regularizer contrast across the instrument’s configuration space, deferred from Section 7.2. Twelve variants: the default funnel at five seeds {42, 1, 7, 123, 2024} and seven single-axis perturbations (branch count ∈ {3, 8}, chain depth ∈ {30, 80}, abandonment ∈ {0.2, 0.5}, sink entropy 0.3). Entries are mean±sd of the interior Spearman across the twelve variants at λ1 = 1; “margin” is graph-Sobolev−Tikhonov, “min” its smallest value over the twelve, and “> 0” the count with positive margin. The contrast holds under every variant; on the topological-sort support the graph-Sobolev ordering is itself configuration-invariant. Reproduced by the released regularizer_robustness experiment. Spearman vs. ϕtrue

E

margin (Sob−Tik)

Support

graph-Sobolev

Tikhonov

mean

min

>0

Topological sort Helmholtz–Hodge

+0.99 ± 0.00 +0.78 ± 0.17

−0.34 ± 0.06 −0.38 ± 0.09

+1.34 ± 0.05 +1.15 ± 0.22

+1.20 +0.45

12/12 12/12

Reproduction

The released repository (https://github.com/MohammadForouhesh/gauge-flow-recovery, with a self-contained README) contains the core solver (poisson_inverse/core.py), the synthetic instrument (poisson_inverse/synthetic.py), the experiments (poisson_inverse/experiments.py), and two scripts: scripts/run_experiments.py writes all numerical results to JSON, and scripts/make_figures.py regenerates Figures 1, 3 and 4. The default seed reproduces every value reported in Tables 1 and 8 and in Sections 5.7 and 7. The real-data results of Section 8 are reproduced by poisson_inverse/loaders.py (the RetailRocket, Trivago, and OTTO loaders, which reduce each corpus to its event-type state space) together with scripts/run_real_data.py (which runs the interpretability, regularizer-contrast, residual, and bootstrap checks) and scripts/make_real_figures.py (which regenerates Figures 2, 5 and 6 from the per-dataset result files). Only the single raw event file per corpus is required; the loaders construct the state space and Dirichlet boundary automatically.

15

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

graph-Sobolev ( 1 = 1)

1.00

Tikhonov ( 1 = 1)

0.75 recovered

0.50 0.25 0.00 0.25 0.50 0.75

0.70

0.75

0.80

0.85

0.90

true (planted absorption prob.)

0.70

0.75

0.80

0.85

0.90

true (planted absorption prob.)

Figure 3: Recovered ϕ̂ versus planted ϕtrue at λ1 = 1 (synthetic instrument). Graph-Sobolev (left) retains a positive relationship with the planted field; Tikhonov (right) collapses every interior state into a flat band near the abandon origin, the range collapse of Proposition 4.

Chain saturation (L = 30)

1.75

harmonic target graph-Sobolev Tikhonov

interior dynamic range

1.50 1.25 1.00 0.75 0.50 0.25 0

10 1 10 2 100 regularization strength 1

101

Figure 4: Interior dynamic range ∆ϕ̂ on a length-30 chain versus λ1 . Graph-Sobolev stays near the unit harmonic target; Tikhonov overshoots at small λ1 then collapses toward zero.

Table 8: Blind sink discovery on the synthetic instrument. The five planted conversion sinks, ranked among 277 states by Poisson residual versus by recovered potential (lower is better); the residual places all sinks in the top seven, invariantly across abandonment regimes. Method Poisson residual (ours) Recovered potential

worst sink rank

regime (abandon ∈ {0.05, 0.2, 0.5})

7 108

invariant n/a

16

Gauge-Invariant, Parameter-Insensitive Regularization on Directed Graphs

RetailRocket (4 states)

Trivago (11 states) clickout item interaction item deals interaction item rating filter selection interaction item info change of sort order search for item search for poi SINK:abandon search for destination interaction item image

SINK:transaction

addtocart

SINK:abandon

view

0.5

0.0 0.5 recovered

1.0

1.5

1.0

0.5 0.0 recovered

0.5

1.0

recovered

Figure 5: Recovered graph-Sobolev potential (λ1 = 1) on RetailRocket and Trivago. Grey bars are pinned sinks; blue bars are recovered interior states. The ordering reproduces the known funnel; the negative interaction item image state on Trivago is discussed in Section 8.1.

1.0 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6

OTTO: recovered potential by event type (105-state multi-sink graph) orders

carts

abandon

clicks states

carts states

Figure 6: OTTO recovered potential by event type (105-state multi-sink graph). Each point is an interior state; black bars are event-type means; dashed lines are the three Dirichlet sink levels. The solve separates carts states (mean +0.26, straddling the cart sink) from clicks states (mean −0.01, near abandon).

17