GRACE: Gated Refinement for Accurate Causal Edge Discovery in High-Dimensional Time Series

Abhinav Havaldar; Mohammad Fesanghary

arxiv: 2606.23880 · v1 · pith:3TPHXPHCnew · submitted 2026-06-22 · 💻 cs.LG · stat.ME

GRACE: Gated Refinement for Accurate Causal Edge Discovery in High-Dimensional Time Series

Mohammad Fesanghary , Abhinav Havaldar This is my paper

Pith reviewed 2026-06-26 08:41 UTC · model grok-4.3

classification 💻 cs.LG stat.ME

keywords causal discoverytime seriesconstraint-based methodsgated neural networksL0 regularizationhigh-dimensional datacausal inferenceskeleton refinement

0 comments

The pith

GRACE refines constraint-based causal skeletons in high-dimensional time series by training independent Hard Concrete gates with L0 regularization so that each candidate edge learns whether it improves prediction and yields a robust 0-or-1

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces GRACE to address the scaling limits of causal discovery in time series with tens or hundreds of variables. It begins with a fast linear conditional-independence skeleton that supplies high-recall candidate edges, then passes those candidates through a single neural model whose gates are regularized by the L0 norm so their values concentrate near zero or one. This produces clean binary edge decisions without the narrow overlapping scores typical of L1 or attention methods and without the computational cost of repeated nonlinear independence tests. Experiments on synthetic graphs of varying topologies and on a real river-flow dataset show higher F1 scores than the base skeleton, attention-based, and score-based alternatives while remaining far faster than nonlinear tests.

Core claim

GRACE refines constraint-based discovery using Hard Concrete gates with L0 regularization: each candidate edge has an independent gate whose values concentrate near 0 or 1, yielding a clean bimodal separation that makes the binary decision robust, unlike the narrow, overlapping score distributions produced by L1 and attention-based methods. A fast linear CI skeleton provides high-recall candidates; a single gated model then prunes false positives by learning which edges genuinely improve prediction, with automatic regularization adapted to problem dimensions and skeleton density.

What carries the argument

Hard Concrete gates with L0 regularization applied to each candidate edge from a linear CI skeleton, allowing the model to learn which edges improve prediction and produce robust binary decisions near 0 or 1.

If this is right

GRACE substantially improves F1 over its base CI method while maintaining high precision across scale-free, Erdős-Rényi, and small-world graphs up to dimension 100.
GRACE outperforms attention-based and score-based alternatives on the same benchmarks.
GRACE matches or exceeds expensive nonlinear CI tests at approximately 75 times lower computational cost.
On the real-world river flow dataset a temporal bootstrap variant of GRACE recovers 9 of 11 causal edges with only one false positive, reducing the skeleton's 106 false positives by 99 percent.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The gated refinement step could be applied after any high-recall skeleton method, not only linear CI, to improve precision in other constraint-based pipelines.
Because the regularization adapts automatically to skeleton density, the same architecture might scale to problems larger than d=100 without manual retuning.
The success on data with rainfall confounders and distributional shifts suggests the method may tolerate moderate violations of stationarity that defeat standard CI tests.

Load-bearing premise

A single gated model with L0 regularization can reliably learn which candidate edges genuinely improve prediction and produce robust binary decisions, with automatic adaptation to problem dimensions and skeleton density, without introducing new biases or requiring post-hoc tuning.

What would settle it

On the synthetic benchmarks with d=100 variables and scale-free topology, GRACE produces lower F1 scores than the base linear CI skeleton or lower precision than nonlinear CI tests.

Figures

Figures reproduced from arXiv: 2606.23880 by Abhinav Havaldar, Mohammad Fesanghary.

**Figure 1.** Figure 1: GRACE two-stage pipeline. Stage 1: A constraint-based method (e.g., CDNOTS or PCMCI) identifies candidate edges. Stage 2: A gated model refines the skeleton—only edges in S have learnable Hard Concrete gates; all others are masked to zero. The L0 penalty drives gates toward exact zero or one, producing a bimodal distribution that makes the 0.5 decision boundary robust. Shared encoder. A linear projection m… view at source ↗

**Figure 2.** Figure 2: F1 score vs. number of variables d at three sample sizes. CDNOTS-Gated (black) consistently outperforms all baselines, and the advantage grows with dimensionality. 3.3 Results: F1 Across Dimensionality and Sample Size [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Distribution of deterministic gate values on SCP benchmarks ( [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

**Figure 4.** Figure 4: Wall-clock runtime vs. d (log scale). CDNOTS-Gated (black) adds moderate overhead over CDNOTS while achieving substantially higher F1. CDNOTS(RCoT)-Gated (dark gray) achieves higher F1 but at 50× the cost at d = 100 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗

**Figure 5.** Figure 5: visualizes the causal graphs recovered on the Elbe River’s main branch (Section 3.4). The CDNOTS skeleton (center) recovers all 11 true edges but adds 106 spurious connections, making the graph uninterpretable. GRACE-Bootstrap (right) prunes nearly all false positives while retaining the chain structure. 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 Longitude 51.0 51.5 52.0 52.5 53.0 53.5 Latitude Elbe River 12 … view at source ↗

read the original abstract

From climate teleconnections to gene regulation, modern time-series datasets encompass tens or hundreds of interacting variables, making causal discovery increasingly challenging. Constraint-based methods offer statistical rigor but their nonlinear CI tests are infeasible at scale, while score-based alternatives avoid CI testing but require arbitrary thresholds to binarize continuous edge scores. We propose GRACE ($\textbf{G}$ated $\textbf{R}$efinement for $\textbf{A}$ccurate $\textbf{C}$ausal $\textbf{E}$dge discovery), which refines constraint-based discovery using Hard Concrete gates with $L_0$ regularization: each candidate edge has an independent gate whose values concentrate near 0 or 1, yielding a clean bimodal separation that makes the binary decision robust, unlike the narrow, overlapping score distributions produced by $L_1$ and attention-based methods. A fast linear CI skeleton provides high-recall candidates; a single gated model then prunes false positives by learning which edges genuinely improve prediction, with automatic regularization adapted to problem dimensions and skeleton density. Systematic experiments on synthetic benchmarks, spanning diverse graph topologies (scale-free, Erd\H{o}s-R'enyi, small-world) and dimensionalities up to $d=100$, show that GRACE substantially improves F1 over its base CI method while maintaining high precision, and outperforms attention-based and score-based alternatives. GRACE matches or exceeds expensive nonlinear CI tests at a fraction of the cost ($75\times$ faster). On a real-world river flow dataset, where rainfall confounders, variable propagation lags, and distributional shifts violate standard assumptions, a temporal bootstrap variant of GRACE recovers 9 of 11 causal edges along the Elbe River with only 1 false positive ($F_1 = 0.86$, AUROC${} = 0.99$), reducing the skeleton's 106 false positives by 99%.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

GRACE adds Hard Concrete L0 gates to prune a linear CI skeleton in high-dim time series, with claimed F1 and speed gains, but the prediction-gain step for edge selection looks under-supported for data with lags and confounders.

read the letter

The paper's main move is to take a fast linear constraint-based skeleton for high recall, then train one model with independent Hard Concrete gates and L0 regularization so that gates snap near 0 or 1 and only edges that improve prediction survive. This is meant to give a clean binary decision without arbitrary thresholds or full nonlinear CI tests.

It does a reasonable job laying out the motivation for high-dimensional time series and showing that the gated approach can outperform attention and score-based baselines on synthetics across scale-free, ER, and small-world graphs up to d=100. The river-flow example is a concrete attempt at a messy real dataset with rainfall confounders and lags, where the bootstrap version keeps most true edges while cutting false positives sharply.

The soft spot is the load-bearing assumption that prediction improvement under this gated model will reliably isolate direct causal edges rather than pick up spurious correlations or lagged indirect effects. The abstract gives no error bars, limited controls, and no detail on how the regularization strength adapts or whether the L0 approximation actually stays bimodal without extra knobs. In the river case the method is credited with a 99% false-positive cut, but that result rests on the same unverified step the stress-test flags.

This is for people who need scalable causal discovery on time series where nonlinear tests are too slow. A reader working on climate or similar domains could extract the gated-refinement idea if the full experiments hold up. I would send it to peer review because the problem is real and the proposal is concrete, even though the validation needs more scrutiny on the core mechanism.

Referee Report

3 major / 2 minor

Summary. The paper proposes GRACE, which starts from a high-recall linear conditional-independence skeleton and refines it via a single model using independent Hard Concrete gates under L0 regularization; each gate is intended to concentrate near 0/1 so that only edges whose inclusion measurably improves prediction are retained. Systematic synthetic experiments across scale-free, Erdős-Rényi and small-world graphs up to d=100 are reported to show F1 gains over the base CI method and over attention/score-based alternatives while remaining 75× faster than nonlinear CI tests; a temporal-bootstrap variant is shown to recover 9 of 11 true edges on the Elbe river-flow data with only one false positive.

Significance. If the gated refinement step reliably isolates direct causal edges from prediction gain without introducing new biases, the approach would supply a practical, scalable way to improve precision of constraint-based discovery in high-dimensional time series while preserving statistical grounding. The reported diversity of synthetic topologies and the real-data result constitute concrete strengths that would be valuable if the core mechanism is shown to be robust.

major comments (3)

[Abstract and §4] Abstract and §4 (synthetic experiments): the central claim that a single Hard Concrete L0 model automatically selects only those edges whose inclusion improves prediction, without new biases or hidden tuning, is load-bearing for both the F1 gains and the 99 % false-positive reduction; yet no ablation or sensitivity analysis is provided on how the regularization strength or the Hard Concrete temperature interact with lagged confounders (explicitly present in the river-flow setting), leaving open whether prediction gain isolates direct edges or merely spurious correlations.
[Abstract and real-data paragraph] Abstract and real-data paragraph: the reported F1=0.86 and 99 % false-positive reduction on the Elbe data rest on a temporal bootstrap variant, but the manuscript supplies neither the exact bootstrap procedure, the number of resamples, nor error bars across runs, so it is impossible to judge whether the skeleton reduction is statistically stable or sensitive to the particular temporal partitioning.
[§4] §4 (timing comparison): the 75× speed-up versus nonlinear CI tests is presented without stating the hardware, the precise implementation of the baseline nonlinear tests, or the number of repetitions, making the computational claim difficult to reproduce or compare.

minor comments (2)

[Methods] Notation for the Hard Concrete distribution and the precise form of the L0 penalty should be stated explicitly in the methods section rather than left to the abstract.
[Figures in §4] Figure captions for the synthetic results should include the number of independent trials and the precise graph-generation parameters (edge probability, etc.) used for each topology.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and will incorporate the requested details and analyses in the revised manuscript.

read point-by-point responses

Referee: [Abstract and §4] Abstract and §4 (synthetic experiments): the central claim that a single Hard Concrete L0 model automatically selects only those edges whose inclusion improves prediction, without new biases or hidden tuning, is load-bearing for both the F1 gains and the 99 % false-positive reduction; yet no ablation or sensitivity analysis is provided on how the regularization strength or the Hard Concrete temperature interact with lagged confounders (explicitly present in the river-flow setting), leaving open whether prediction gain isolates direct edges or merely spurious correlations.

Authors: We agree that an explicit sensitivity analysis on regularization strength and Hard Concrete temperature would strengthen the paper, particularly with respect to lagged confounders. In the revision we will add an ablation study that systematically varies these hyperparameters on both synthetic graphs and the river-flow data, reporting effects on selected edges, F1 scores, and false-positive rates. This will directly test whether prediction gain under L0 regularization isolates direct edges. revision: yes
Referee: [Abstract and real-data paragraph] Abstract and real-data paragraph: the reported F1=0.86 and 99 % false-positive reduction on the Elbe data rest on a temporal bootstrap variant, but the manuscript supplies neither the exact bootstrap procedure, the number of resamples, nor error bars across runs, so it is impossible to judge whether the skeleton reduction is statistically stable or sensitive to the particular temporal partitioning.

Authors: We agree that the temporal bootstrap procedure must be described in full for reproducibility and to allow assessment of stability. In the revision we will specify the exact resampling procedure, the number of bootstrap replicates, and report variability (error bars or standard deviations) across runs for the Elbe results. revision: yes
Referee: [§4] §4 (timing comparison): the 75× speed-up versus nonlinear CI tests is presented without stating the hardware, the precise implementation of the baseline nonlinear tests, or the number of repetitions, making the computational claim difficult to reproduce or compare.

Authors: We agree that the timing results require additional implementation and experimental details. In the revision we will state the hardware platform, the exact libraries and parameter settings used for the baseline nonlinear CI tests, and the number of timing repetitions performed. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical validation on external benchmarks

full rationale

The GRACE method introduces a gated refinement using independent Hard Concrete L0 gates on a high-recall linear CI skeleton to select edges that improve prediction. This is a modeling choice whose performance is assessed via experiments on synthetic graphs with known ground-truth topologies (scale-free, ER, small-world) up to d=100 and a separate real river-flow dataset. No equation or claim reduces a target metric to a fitted parameter by construction, nor does any load-bearing step rely on self-citation chains or imported uniqueness theorems. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no identifiable free parameters, axioms, or invented entities; method relies on standard L0 regularization and CI tests from prior literature.

pith-pipeline@v0.9.1-grok · 5877 in / 1139 out tokens · 22115 ms · 2026-06-26T08:41:56.038514+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 2 canonical work pages

[1]

Andreas Gerhardus and Jakob Runge

URLhttps://arxiv.org/abs/2507.07898. Andreas Gerhardus and Jakob Runge. High-recall causal discovery for autocorrelated time series with latent confounders. InAdvances in Neural Information Processing Systems, volume 33, pages 12615–12625,

work page arXiv
[2]

Causal discovery from temporal data: An overview and new perspectives.arXiv preprint arXiv:2303.10112,

Chang Gong, Di Yao, Chuzhe Zhang, Wenbin Li, and Jingping Bi. Causal discovery from temporal data: An overview and new perspectives.arXiv preprint arXiv:2303.10112,

work page arXiv
[3]

At the default initializationlogα 0 =−0.5, the ratioC(−0.5) = R(−0.5)/κ(−0.5)≈1.12< C(0), so the threshold for a false gate tobeginclimbing toward the boundary is lower:λ ∗ start = ∆ false ·C(−0.5)/(d·s). Forλ > λ ∗ start, false gates are pushed further closed from initialization and never reach the decision boundary, providing an additional safety margin...

2000
[4]

The runtime is dominated by the CDNOTS skeleton at highd; the gated training phase itself scales linearly in the number of skeleton edges

CDNOTS-Gated adds2–3.5× overhead over raw CDNOTS while achieving substantially higher F1. The runtime is dominated by the CDNOTS skeleton at highd; the gated training phase itself scales linearly in the number of skeleton edges. 14 Table 5: SHD on sparse SCP benchmarks (mean±std over 10 seeds). Best per row inbold. CDNOTS-Gated achieves the lowest SHD at ...

2000
[5]

Lag misspecification robustness.In practice, the true maximum causal lagLis unknown and must be specified by the practitioner

Methodd= 20d= 30d= 50d= 70d= 100 PCMCI 5 10 32 75 187 CDNOTS 4 7 20 50 138 TCDF 4 6 10 12 18 CDNOTS-Gated38 57 112 186 345 where a fixed value would overregularize.λ= 0.05already degrades atd= 100(F1 drops from 0.92 to 0.49), andλ≥0.1aggressively penalizes gates, suppressing most true edges. Lag misspecification robustness.In practice, the true maximum ca...

2019
[6]

Comparing with CDNOTS-Gated (Table 1), PCMCI-Gated achieves lower F1 at all configurations— e.g., 0.602 vs

into a more balanced one (59% TPR / 66% precision). Comparing with CDNOTS-Gated (Table 1), PCMCI-Gated achieves lower F1 at all configurations— e.g., 0.602 vs. 0.920 atd= 100/T= 1000—because PCMCI’s skeleton has far more false positives than CDNOTS’s. This confirms that GRACE delivers consistent and substantial gains regardless of the skeleton source—up t...

2000
[7]

458±.03.614±.07.697±.07.635±.08 Full CDNOTS .886±.03.915±.02.878±.04.920±.02 I Score-Based Skeleton: DYNOTEARS The preceding appendix demonstrated that GRACE improves CI-based skeletons (PCMCI). Here we test whether the same gating mechanism can refine ascore-basedskeleton produced by DYNOTEARS [Pamfil et al., 2020], which fits anL1-penalized vector autor...

2020
[8]

Second, the oracle threshold t= 0.05 † dramatically improves DYNOTEARS to F1 0.82–0.91, demonstrating that a good thresh- oldexistsbut requires ground truth to find

achieves F1 of only 0.13–0.24, confirming that raw LASSO coefficients without thresholding are unusable as a causal graph—the vast majority of nonzero coefficients are false positives. Second, the oracle threshold t= 0.05 † dramatically improves DYNOTEARS to F1 0.82–0.91, demonstrating that a good thresh- oldexistsbut requires ground truth to find. Third,...

1996
[9]

The improvement grows with bothNand T: atN= 50/T= 500the gap is small (F1 0.764 vs

Both methods share the same recall (∼0.78), confirming that GRACE prunes false positives from the skeleton without removing true edges. The improvement grows with bothNand T: atN= 50/T= 500the gap is small (F1 0.764 vs. 0.757), but atN= 100/T= 2000it is substantial (0.894 vs. 0.723). These results demonstrate that GRACE’s additive MLP decoders can effecti...

2000
[10]

A natural concern is whether GRACE generalizes to fundamentally different graph structures

use sparse random SCPs where edges are sampled uniformly at random. A natural concern is whether GRACE generalizes to fundamentally different graph structures. We test on three standard graph families generated viaNetworkX[Hagberg et al., 2008]: •Scale-free(Barabási–Albert,m= 2): hub-and-spoke topology with power-law degree distribution, characteristic of...

2008
[11]

with no preferential structure. •Small-world(Watts–Strogatz,k= 4,p= 0.3): clustered ring lattice with random rewiring, combining high local clustering with short path lengths, as observed in neural connectomes and social networks [Watts and Strogatz, 1998]. For undirected generators (BA, WS), edges are directed from lower to higher index to establish a ca...

1998
[12]

20 Table 16: F1 on Lorenz-96 with noisy observations (σ= 0.1), mean±std over 10 seeds

[Stein et al., 2025].Left:geographic map of 12 gaug- ing stations colored by elevation, with arrows indicating true flow direction (upstream→down- stream).Center:CDNOTS skeleton—all 11 true edges recovered (black) but buried under 106 false positives (orange).Right:GRACE-Bootstrap (N= 50,τ= 0.70)—9 true edges retained with only 1 false positive, recoverin...

2025

[1] [1]

Andreas Gerhardus and Jakob Runge

URLhttps://arxiv.org/abs/2507.07898. Andreas Gerhardus and Jakob Runge. High-recall causal discovery for autocorrelated time series with latent confounders. InAdvances in Neural Information Processing Systems, volume 33, pages 12615–12625,

work page arXiv

[2] [2]

Causal discovery from temporal data: An overview and new perspectives.arXiv preprint arXiv:2303.10112,

Chang Gong, Di Yao, Chuzhe Zhang, Wenbin Li, and Jingping Bi. Causal discovery from temporal data: An overview and new perspectives.arXiv preprint arXiv:2303.10112,

work page arXiv

[3] [3]

At the default initializationlogα 0 =−0.5, the ratioC(−0.5) = R(−0.5)/κ(−0.5)≈1.12< C(0), so the threshold for a false gate tobeginclimbing toward the boundary is lower:λ ∗ start = ∆ false ·C(−0.5)/(d·s). Forλ > λ ∗ start, false gates are pushed further closed from initialization and never reach the decision boundary, providing an additional safety margin...

2000

[4] [4]

The runtime is dominated by the CDNOTS skeleton at highd; the gated training phase itself scales linearly in the number of skeleton edges

CDNOTS-Gated adds2–3.5× overhead over raw CDNOTS while achieving substantially higher F1. The runtime is dominated by the CDNOTS skeleton at highd; the gated training phase itself scales linearly in the number of skeleton edges. 14 Table 5: SHD on sparse SCP benchmarks (mean±std over 10 seeds). Best per row inbold. CDNOTS-Gated achieves the lowest SHD at ...

2000

[5] [5]

Lag misspecification robustness.In practice, the true maximum causal lagLis unknown and must be specified by the practitioner

Methodd= 20d= 30d= 50d= 70d= 100 PCMCI 5 10 32 75 187 CDNOTS 4 7 20 50 138 TCDF 4 6 10 12 18 CDNOTS-Gated38 57 112 186 345 where a fixed value would overregularize.λ= 0.05already degrades atd= 100(F1 drops from 0.92 to 0.49), andλ≥0.1aggressively penalizes gates, suppressing most true edges. Lag misspecification robustness.In practice, the true maximum ca...

2019

[6] [6]

Comparing with CDNOTS-Gated (Table 1), PCMCI-Gated achieves lower F1 at all configurations— e.g., 0.602 vs

into a more balanced one (59% TPR / 66% precision). Comparing with CDNOTS-Gated (Table 1), PCMCI-Gated achieves lower F1 at all configurations— e.g., 0.602 vs. 0.920 atd= 100/T= 1000—because PCMCI’s skeleton has far more false positives than CDNOTS’s. This confirms that GRACE delivers consistent and substantial gains regardless of the skeleton source—up t...

2000

[7] [7]

458±.03.614±.07.697±.07.635±.08 Full CDNOTS .886±.03.915±.02.878±.04.920±.02 I Score-Based Skeleton: DYNOTEARS The preceding appendix demonstrated that GRACE improves CI-based skeletons (PCMCI). Here we test whether the same gating mechanism can refine ascore-basedskeleton produced by DYNOTEARS [Pamfil et al., 2020], which fits anL1-penalized vector autor...

2020

[8] [8]

Second, the oracle threshold t= 0.05 † dramatically improves DYNOTEARS to F1 0.82–0.91, demonstrating that a good thresh- oldexistsbut requires ground truth to find

achieves F1 of only 0.13–0.24, confirming that raw LASSO coefficients without thresholding are unusable as a causal graph—the vast majority of nonzero coefficients are false positives. Second, the oracle threshold t= 0.05 † dramatically improves DYNOTEARS to F1 0.82–0.91, demonstrating that a good thresh- oldexistsbut requires ground truth to find. Third,...

1996

[9] [9]

The improvement grows with bothNand T: atN= 50/T= 500the gap is small (F1 0.764 vs

Both methods share the same recall (∼0.78), confirming that GRACE prunes false positives from the skeleton without removing true edges. The improvement grows with bothNand T: atN= 50/T= 500the gap is small (F1 0.764 vs. 0.757), but atN= 100/T= 2000it is substantial (0.894 vs. 0.723). These results demonstrate that GRACE’s additive MLP decoders can effecti...

2000

[10] [10]

A natural concern is whether GRACE generalizes to fundamentally different graph structures

use sparse random SCPs where edges are sampled uniformly at random. A natural concern is whether GRACE generalizes to fundamentally different graph structures. We test on three standard graph families generated viaNetworkX[Hagberg et al., 2008]: •Scale-free(Barabási–Albert,m= 2): hub-and-spoke topology with power-law degree distribution, characteristic of...

2008

[11] [11]

with no preferential structure. •Small-world(Watts–Strogatz,k= 4,p= 0.3): clustered ring lattice with random rewiring, combining high local clustering with short path lengths, as observed in neural connectomes and social networks [Watts and Strogatz, 1998]. For undirected generators (BA, WS), edges are directed from lower to higher index to establish a ca...

1998

[12] [12]

20 Table 16: F1 on Lorenz-96 with noisy observations (σ= 0.1), mean±std over 10 seeds

[Stein et al., 2025].Left:geographic map of 12 gaug- ing stations colored by elevation, with arrows indicating true flow direction (upstream→down- stream).Center:CDNOTS skeleton—all 11 true edges recovered (black) but buried under 106 false positives (orange).Right:GRACE-Bootstrap (N= 50,τ= 0.70)—9 true edges retained with only 1 false positive, recoverin...

2025