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Graph Anomaly Detection as Finite-Horizon Control: Training-Free Scoring via Empirical Bayes

Fred Xu, Thomas Markovich, Florence Regol, Yizhou Sun

arXiv:2609.38424v1cs.LG

TL;DR

Existing graph anomaly detectors tie graph trust, spectral weighting, and scoring to model choices, complicating label-free detection across graph regimes. EB-GAD fits a graph-aware prior and selects among shared-prior finite-horizon and equilibrium scores without labels. It is best or tied-best on 9 of 11 benchmarks, while its selector cutoffs are not claimed to transfer to other graph families.

  • Problem

    Existing GAD methods tie graph trust, graph-spectral weighting, and anomaly scoring to architectures and tuned objectives, although unlabeled edges vary in how reliably they indicate normality.

    Source: §1 · pp. 1–2

  • Method

    EB-GAD fits a graph-aware normality prior by residual-field empirical Bayes and uses a label-free selector to choose among shared-prior finite-horizon and equilibrium score families.

    Source: §1 · p. 2

  • Results

    On 11 benchmarks, EB-GAD has the best or tied-best AUROC on 9, with margins up to 21.7 points, and is second on BlogCatalog and ACM.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Finite-horizon aggregation or normalization adds 3.6 to 4.7 AUROC points on three datasets and is at parity elsewhere, while equilibrium Mahalanobis scoring remains a limiting case.

    Source: §5 · p. 10

    The selector cutoffs are implementation defaults developed on this paper’s benchmarks, and the authors do not claim they transfer to other graph families.

    Source: §5 · p. 10

Abstract

from arXiv · show

Node-level graph anomaly detection (GAD) identifies nodes whose attributes and interactions deviate from dominant graph regularities. Existing GAD models encode normality and anomaly scoring indirectly through architectures, message passing, reconstruction or contrastive objectives, and tuned score families. This entangles graph trust (how strongly graph structure should define normality), graph-spectral weighting, and anomaly-score choice, yielding scores that are costly, opaque, and unstable across graph regimes. We propose EB-GAD (Empirical-Bayes GAD), a training-free framework that models normality as graph-aware generalized Ornstein-Uhlenbeck (GOU) relaxation toward a graph-filtered template. Empirical Bayes fits the graph precision from the residual-field likelihood; the GOU then turns scoring into a closed-form finite-horizon control energy, the minimum effort to steer a feature-neutral node to its observed endpoint along graph-spectral relaxation. Sweeping relaxation horizon and endpoint tolerance yields a bank of scores that share one fitted prior: equilibrium Mahalanobis scoring is one limit, while finite-horizon control-energy and scale-normalized ratio scores reveal anomalies that static equilibrium scoring can mask. A label-free selector chooses the score family from feature homophily, edge density, and feature dimension, then ranks candidates by fitted-null deviation and rank stability. On 11 benchmarks and without labels at any step, EB-GAD has the best or tied-best AUROC on 9: the four financial fraud networks (up to 3.7M nodes), the YelpChi and Amazon review graphs, Weibo, Reddit and Facebook, with margins of up to 21.7 points. It is second on BlogCatalog and ACM.

1 Introduction

Graph anomaly detection must distinguish abnormal attributes and neighborhoods despite unlabeled, variably trustworthy edges, while existing methods often tie graph trust, spectral weighting, and scoring to model choices. EB-GAD makes these choices explicit through an empirically fitted graph prior and a finite-horizon score bank; without labels, it is best or tied-best on 9 of 11 benchmarks.

1 Introduction

  • Existing methods encode graph trust, spectral weighting, and anomaly scoring indirectly, making scores hard to interpret and unstable across graph regimes.Several baselines score below chance on Elliptic and Facebook in native-score evaluation.Source: §1 · pp. 1–2
  • EB-GAD fits a graph-aware normality prior by empirical Bayes and scores nodes through finite-horizon control energy, with one fitted prior shared across a bank of scores.Varying horizon and endpoint tolerance probes short-horizon effort, equilibrium roughness, and scale-normalized roughness.Source: §1 · p. 2
  • The framework is training-free in that it uses no learned neural parameters, gradient descent, labels, or validation sets, but it still fits three scalars and uses a fixed label-free selector.The selector chooses among equilibrium, control-energy, control-ratio, and profile scores; training-free does not mean hyperparameter-free.Source: §1 · p. 2
  • On 11 benchmarks, EB-GAD has the best or tied-best AUROC on 9, with margins up to 21.7 points; it is second on BlogCatalog and ACM.The leading results include four financial fraud networks, YelpChi, Amazon, Weibo, Reddit, and Facebook.Source: §1 · p. 2

2 Background

The background frames graph anomaly detection as ranking atypical nodes while choosing how much graph structure should define normality. It introduces empirical Bayes as a way to fit normal-data model parameters and generalized Ornstein–Uhlenbeck dynamics as a tractable graph-aware process.

2.1 Graph Anomaly Detection

  • Node-level graph anomaly detection ranks nodes whose attributes and structural interactions depart from graph regularities, across settings including fraud, camouflage, and heterophily.Existing method families include reconstruction or generative, self-supervised or contrastive, and one-class approaches; EB-GAD instead selects linear-Gaussian model parameters from unlabeled data.Source: §2.1 · p. 2

2.2 Empirical Bayes for Anomaly Detection

  • Empirical Bayes selects normality-model hyperparameters by maximizing marginal likelihood, which integrates latent variables when present and balances fit against complexity.Here it is applied to the residual field from a graph-filtered template, with a graph-coupled likelihood and nodewise scores from the same joint Gaussian energy.Source: §2.2 · p. 3

2.3 Generalized Ornstein–Uhlenbeck Dynamics

  • A generalized Ornstein–Uhlenbeck process supplies a linear-Gaussian model whose stationary mean can be set to a template and whose graph-aware precision supports closed-form transitions.Its stationary law is Gaussian, and the graph enters both the stationary mean and covariance while preserving closed-form fitting and scoring.Source: §2.3 · pp. 3–4

3 Graph Anomaly Detection via Finite-Horizon Control

EB-GAD composes a graph Gaussian prior, generalized Ornstein–Uhlenbeck dynamics, and quadratic scores into a label-free anomaly-scoring framework. It fits the prior from residuals and uses a shared-prior bank of finite-horizon and equilibrium scores, followed by a fixed selector.

3 Graph Anomaly Detection via Finite-Horizon Control

  • EB-GAD combines standard graph-Gaussian priors, OU dynamics, and quadratic scores, with empirical-Bayes calibration and label-free score selection as the contribution.Its GOU is a representational device for static data, not a claim that the observed data evolved through OU dynamics.Source: §3 · p. 4, §3.1 · p. 4

3.1 Graph-Aware GOU Process

  • The graph-aware GOU relaxes toward a low-pass graph-filtered feature template, with precision blending graph trust and Matérn inverse length-scale to set graph-frequency weighting.The precision parameters distinguish overall trust in graph structure from the frequency at which precision shifts from a low-frequency floor to spectral growth.Source: §3.1 · pp. 4–5

3.2 Finite-Horizon Generative Model and Control Energy

  • Finite-horizon Gaussian endpoints define a control-energy score: the minimum effort to steer a feature-neutral start to observed features under attraction toward the template.The score is the negative log-density of the unconditioned endpoint distribution up to a constant; spectrally rough deviations are more expensive, while smooth ones are cheaper.Source: §3.2 · pp. 5–6

3.3 Per-Node Anomaly Scoring

  • Varying horizon and endpoint tolerance yields nodewise control-energy and scale-normalized ratio scores, whose equilibrium limits are graph-Mahalanobis scores.The ratio addresses cases where raw residual magnitude can make a high-magnitude normal node appear anomalous; its numerator measures retained-subspace energy while its denominator uses the full nodewise residual norm.Source: §3.3 · p. 7

3.4 Empirical Bayes Prior Selection

  • Empirical Bayes fits the residual-space likelihood for candidate templates and reuses the fitted precision across finite-horizon scores; a fixed label-free rule selects a score family and ranks candidates by fitted-null quality and neighboring-rank stability.The selector uses graph statistics including feature homophily, edge density, feature dimension, and truncation; its cutoffs are implementation defaults rather than a single derived principle.Source: §3.4 · pp. 7–9

4 Experiments

Across 11 benchmarks, EB-GAD achieves the best or tied-best AUROC on 9, while controlled comparisons examine finite-horizon scoring, selector robustness, and runtime. Its results are reported under a fixed, label-free evaluation procedure.

Setup

  • The evaluation uses one fixed EB-GAD procedure across 11 benchmarks, with no labels entering before evaluation and the same node set scored across methods.The reported benchmarks include seven standard GAD graphs and four financial networks, reaching 3.7M nodes.Source: §4 · p. 9

Main results

  • EB-GAD has the best or tied-best AUROC on 9 of 11 datasets, leading on five datasets by 2.6 to 21.7 points and tying TAM on Facebook at 91.4.It is second on BlogCatalog and ACM; the Weibo and Reddit entries depend on the anchor bandwidth rule.Source: §4 · p. 9, Table 1 · p. 10

Finite-horizon contribution

  • Three finite-horizon selections gain 3.6 to 4.7 AUROC points over the same-configuration equilibrium score, while two control-energy fusions are near parity.A single finite-horizon score without tolerance grids, fusion, or aggregation is within 0.3 points of its equilibrium limit or below it.Source: §4 · p. 9, Table 2 · p. 10

Selector and robustness

  • Perturbing selector cutoffs by ±25% leaves selection unchanged in 99.4% of evaluations, while quality-only or stability-only selection fails on Facebook.The combined selector gives 91.4 AUROC on Facebook; robustness tests report losses under contamination and heavy-tailed residuals, but larger losses for multimodal normals and failure with binarized features.Source: §4 · p. 9

Encoders and scalability

  • Replacing PCA with trained encoders leaves the training-free pipeline ahead on 8 of 11 datasets; a complete EB-GAD run ranges from 8 seconds to 33 minutes on 16 CPU threads.On DGraph, 30 of the 33 minutes are spent on eigendecomposition; the two trained graph baselines fitting in memory take 65 and 69 minutes on a GPU.Source: §4 · p. 10

5 Scope, Limitations and Conclusion

EB-GAD combines a graph-aware GOU prior, empirical-Bayes fitting, and label-free score selection; its equilibrium score is one limit of a shared finite-horizon score bank. The paper reports strong benchmark results while identifying boundaries involving selector transfer, data assumptions, graph operators, and deployment fairness.

5 Scope, Limitations and Conclusion

  • EB-GAD uses a graph-aware GOU prior and residual-space likelihood to fit graph precision, then selects among four score families without labels.The equilibrium Mahalanobis score is one limit; aggregating or normalizing finite-horizon scores adds 3.6 to 4.7 AUROC points on three datasets and is at parity elsewhere.Source: §5 · p. 10
  • The selector cutoffs are benchmark-developed implementation defaults, and the authors do not claim they transfer to other graph families.Other stated boundaries include Gaussian residual assumptions, unstudied large contiguous anomalous regions, and possible degree or activity bias in user-graph scores that should be audited across user segments.Source: §5 · p. 10

A Derivation of Technical Results

A.1 Modeling Details

The modeling details define a graph-coupled Gaussian residual prior and connect its finite-horizon transition to control-energy scoring. They also specify the modeled subspace, equilibrium limit, and fitting considerations.

Modeling details

  • Row-major vectorization lets the graph precision couple nodes while acting trivially across feature dimensions; the low-pass template’s bandwidth also controls its low-frequency scale.The same residual operator is used for likelihood evaluation and scoring.Source: §A.1 · p. 15
  • The likelihood is concave in graph trust ρ, but κ and γ are searched on bounded grids because their profile objectives are non-concave or non-convex.The squared-residual fit can be sensitive to extreme anomalies, though the paper notes that low anomaly fractions and spectral truncation mitigate this.Source: §A.1 · p. 15
  • The finite-horizon GOU endpoint has a Gaussian transition, and its control energy is a per-node contribution to total energy because graph precision couples nodes.Endpoint tolerance acts as an inverse variance; under hard endpoint matching, the quadratic form uses the inverse endpoint covariance.Source: §A.1 · p. 15
  • The equilibrium covariance is the infinite-horizon limit of finite-horizon covariance, and finite-horizon precision converges exponentially toward equilibrium precision.For a fixed residual, the corresponding likelihood objectives converge as the horizon grows; at three multiples of the inverse minimum precision, the uniform relative precision error is below 0.5%.Source: §A.1 · pp. 16–18
  • Likelihood and energy use the retained spectral projection, while the ratio score divides projected energy by the full nodewise residual norm.Disconnected-graph null modes are not removed explicitly; retained modes instead must have positive precision, with the isotropic part handling graph-uninformative directions.Source: §A.1 · pp. 15, 17

B.1 Dataset Specifications

The benchmark suite spans social, review, academic, and financial graphs, with anomaly labels tied to dataset-specific entities and tasks. The appendix also describes the comparison baselines.

B.1 Dataset Specifications

  • The 11 benchmark graphs cover social and review networks, academic graphs, and financial transaction or lending networks, including DGraph with about 3.7M nodes.Tasks include suspicious-account, spam-review, injected contextual or structural anomaly, and fraudulent-actor detection.Source: §B.1 · p. 18
  • Table 3 reports raw benchmark statistics, with feature dimension and anomaly rate identified as #Feat and Anom.%.Source: Table 3 · p. 19

B.2 Baseline Models

  • The compared baselines include density-based LOF and DIF, graph reconstruction and autoencoder methods, contrastive methods, local-affinity TAM, and diffusion-based DiffGAD.The appendix names these approaches and describes their scoring or representation strategies.Source: §B.2 · p. 19

B.3 Native-Score Evaluation and Orientation Audit

The comparison evaluates baselines in their fixed, native anomaly-score directions; reversing scores after seeing labels is treated only as an oracle diagnostic. Audited sub-50 fraud-baseline AUROCs reflect genuine anticorrelation, not missing data.

B.3 Native-Score Evaluation and Orientation Audit

  • Native-score AUROC is the only baseline result used in the main table; label-informed score reversal is excluded from the comparison.The audit uses reversal only diagnostically, not as a competing baseline.Source: §B.3 · p. 19, Table 4 · p. 20
  • In every audited case, normal nodes receive larger native scores on average than anomalous nodes, confirming genuine anticorrelation in the sub-50 fraud-baseline AUROCs.The negated-score AUROC is an oracle value reported only to show the effect of reversal.Source: §B.3 · pp. 19–20, Table 4 · p. 20

B.4 Selector and Hyperparameter Specification

The selector uses unlabeled graph statistics and fitted-null diagnostics to choose among score families, with fixed implementation defaults and a shared EB fit. The appendix also reports equilibrium-anchor comparisons, scalability measures, and runtime caveats.

B.4 Selector and Hyperparameter Specification

  • Selector cutoffs are implementation defaults fixed during development on the benchmark families, not derived quantities, and the same cutoffs are used across all 11 datasets.The appendix classifies the constants and reports cutoff perturbations separately.Source: §B.4 · p. 20
  • The run uses z-scored features, PCA for equilibrium and profile families when Dx > 100, and a likelihood-selected bandwidth for finite-horizon families.The equilibrium anchor instead uses the grid bandwidth nearest γ0; graph spectra are truncated for larger graphs.Source: §B.4 · p. 20
  • The ordered selector routes among equilibrium, control-energy, control-ratio, and profile families using homophily, edge density, feature dimension, and unlabeled diagnostics.NullKS, Berk–Jones, mixture-mass, and branch-specific criteria determine whether a rule applies; within-family candidates are filtered and ranked by score quality and stability.Source: §B.4 · p. 21

B.5 Equilibrium Scores

  • Facebook’s equilibrium scores differ sharply: NullKS selects the score with AUROC 47.4 rather than 89.7, illustrating the cost of the selector’s choice there.Across datasets, NullKS selects the better equilibrium score on Weibo, Amazon, BlogCatalog, and ACM, but the worse one on Reddit, Facebook, and T-Finance.Source: §B.5 · p. 23, Table 8 · p. 23

B.6 Resource Utility and Scalability Measures

  • For graphs with at least 20K nodes, truncated eigendecomposition retains low-frequency modes and reduces eigenvector-matrix memory from O(n^2) to O(nk).Graph-filter and prior operations use eigenspace products, and large-graph scoring uses the eigen cache rather than a dense precision matrix.Source: §B.6 · p. 24

B.7 Computation Time

  • EB-GAD runs take 8 seconds on Facebook and 33 minutes on DGraph, but these times are not a controlled comparison because the methods use different hardware.The DGraph eigendecomposition accounts for 30 minutes; candidate-pool datasets take 11–26 minutes, with repeated eigendecompositions adding overhead.Source: §B.7 · p. 24, Table 9 · p. 24

B.8 Dimensionality Reduction and Nonlinear Extensions

EB-GAD uses linear PCA to preserve its closed-form Gaussian scoring, while nonlinear encoders require a variational extension. In the reported comparison, training-free EB-GAD leads on 8 of 11 datasets, with trained encoders ahead on three.

B.8 Dimensionality Reduction and Nonlinear Extensions

  • PCA preserves Gaussianity, so the closed-form anomaly scores and residual-space likelihood apply exactly in the projected space.The PCA map is deterministic and unsupervised and retains components by variance.Source: §B.8 · p. 25
  • A nonlinear encoder makes latent features non-Gaussian, removing the closed-form residual-space likelihood and requiring a variational bound for prior selection.The proposed bridge-based ELBO is described only as a future extension and is not used in the main pipeline.Source: §B.8 · p. 25
  • Kernel PCA, random Fourier features, and GPLVM span a trade-off from closed-form tractability toward expressive but variational methods; further exploration is left to future work.Kernel PCA and random Fourier features are training-free but make the Gaussian likelihood approximate; GPLVM requires variational optimization.Source: §B.8 · p. 25
  • Training-free EB-GAD is ahead on 8 of 11 datasets by 1.0–22.3 points, while trained encoders lead on Weibo, Facebook, and T-Finance by 0.6, 4.0, and 0.1 points, respectively.The comparison uses the better of two trained encoders per dataset, followed by the same score formula.Source: §B.8 · p. 25, Table 10 · p. 26
  • Where trained encoders hurt, plausible causes include breaking Gaussian structure, distorting graph-frequency relations, and adding seed-dependent variance.The comparison reports that encoders help on two social graphs and hurt on five of six fraud graphs, ACM, BlogCatalog, and Reddit.Source: §B.8 · pp. 25–26

C Regenerated Results and Additional Studies

  • The appendix’s finite-horizon-bank studies use the EB-fitted configuration at γ⋆, and also report the anchor-bandwidth result when γa differs.The studies include isolation, contamination, misspecification, and neighboring-anomaly analyses.Source: C Regenerated Results and Additional… · p. 26

C.1 Protocol, Relabeling Gate and Changes from the Submitted Version

The regenerated protocol fixes the unlabeled pipeline and checks sensitivity to node relabeling and modeling conventions. The appendix also explains a YelpChi ordering artifact and the nonuniform effect of the anchor-bandwidth rule.

C.1 Protocol, Relabeling Gate and Changes from the Submitted Version

  • All reported scores use one label-free protocol with double-precision eigendecompositions, all computed modes retained, and average ranks for ties.The released protocol and regeneration scripts accompany the code.Source: §C.1 · pp. 26–27
  • On full-spectrum datasets, relabeling preserves the selected score with rank correlation at least 0.9994 and AUROC change at most 0.03; truncated YelpChi varies from 70.7 to 72.7.For YelpChi, all 500 retained modes lie in the Laplacian nullspace and the basis is not unique; truncated-dataset results are means over five node orders.Source: §C.1 · p. 26, Table 11 · p. 27
  • YelpChi’s submitted profile result of 95.3 fell to 71.7 after tie-averaging removed node-index dependence; the corrected selector routes it to the equilibrium anchor.In the released order, node index alone has AUROC 94.4; with averaged ties the profile score is 67.3 there and 66.8 after random relabeling.Source: §C.1 · pp. 26–27, Table 12 · p. 28
  • Using the likelihood-selected bandwidth for the equilibrium anchor gave 92.5 on Weibo and 48.5 on Reddit, while the current anchor uses the grid value nearest γ0.The text distinguishes the bandwidth that best explains residual feature energy from the separate question of how well one residual energy ranks anomalies.Source: §C.1 · pp. 27–28
  • The anchor-bandwidth rule was adopted after both bandwidths were evaluated on labels, and its effect is uneven: it gains 12.1 points on Reddit but loses 1.2 on Elliptic++.The anchor bandwidth is γa, the grid value nearest γ0, rather than a fitted quantity; the rule applies only to the anchor.Source: §C.1 · p. 28
  • Excluding Laplacian null modes changes full-spectrum dataset entries by −6.6 to +0.1 points, including BlogCatalog’s 78.6 →72.0 and ACM’s 86.0 →82.1.Choosing every bandwidth by likelihood with the residual-map Jacobian instead leaves Reddit at 61.9 while lowering four other named datasets by 1.6–5.6 points.Source: §C.1 · p. 28

C.2 AUPRC

EB-GAD leads several AUPRC comparisons, while its advantage varies across datasets. In matched spectral-family comparisons, the GOU bank’s strongest gains are on three financial graphs and are attributed to label-free selection rather than uniformly stronger best scores.

C.2 AUPRC

  • Weibo AUPRC is 34.9 against 34.6 for the best baseline; Reddit is 4.5, tied with TAM and CoLA, but only 1.2 points above its base rate.Reddit has 3.3% anomalies, and the passage characterizes its column as unsolved.Source: §C.2 · pp. 28–30, Table 15 · p. 29
  • EB-GAD ranks second on Amazon, third on BlogCatalog and ACM, and fourth on Facebook AUPRC; its Facebook AUROC ties the best at 91.4 while AUPRC 17.2 trails TAM at 22.3.Table 15 reports the AUPRC comparison.Source: §C.2 · p. 28, Table 15 · p. 29
  • EB-GAD leads five fraud AUPRC columns: YelpChi 9.7 vs 7.8, Elliptic 22.1 vs 15.3, Elliptic++ 19.1 vs 12.2, DGraph 2.3 vs 1.9, and T-Finance 53.0 vs 6.6 against the best baseline.The comparisons are reported in Table 15.Source: §C.2 · p. 28, Table 15 · p. 29

C.3 Isolation of the Spectral Precision Family

  • On T-Finance, EB fitting adds 23.6 points over fixed GOU precision; on Elliptic and Elliptic++, all five families’ best members are within 1.2 points, but generic selection falls to 58–62.Elsewhere no family is consistently better: polynomial selection leads on Reddit and Amazon, and heat-kernel selection leads on Facebook.Source: §C.3 · p. 30
  • With identical label-free selection, the GOU bank leads the best generic family by 12.6 points on Elliptic, 11.6 on Elliptic++, and 10.2 on T-Finance.Table 16 fixes the operator, template, PCA, truncation, residuals, and selection, changing only spectral precision.Source: Table 16 · p. 30, §C.3 · p. 30

C.4 Horizon Axis: Matched Single-Score Comparison

Matched comparisons find near-parity between individual finite-horizon scores and their equilibrium limits. The gains reported for selected scores instead come from aggregating or fusing the family, whose horizons share one fitted null.

C.4 Horizon Axis: Matched Single-Score Comparison

  • No single finite-horizon score reproduces the reported gains of +3.6 on Facebook and +4.7 on ACM and T-Finance.The reported gains come from horizon p-value aggregation on Facebook and T-Finance, and scale normalization with stability-ranked fusion on ACM.Source: §C.4 · p. 30
  • A shared fitted prior gives every horizon one plug-in null, making the score family comparable and aggregatable without labels; equilibrium is only one member.The equilibrium limit itself cannot be aggregated across horizons.Source: §C.4 · pp. 30–31
  • The best finite-horizon AUROC is within 0.3 points of equilibrium on nine of ten datasets and 1.4 below it on T-Finance; NullKS-picked scores are within 0.2 on eight datasets.Table 17 compares one finite score with its equilibrium counterpart at a shared configuration, without tolerance grids, fusion, or aggregation.Source: §C.4 · p. 30, Table 17 · p. 31

C.5 Selector Studies

The selector is usually stable to cutoff perturbations, though changed selections can sharply reduce AUROC. Within-family quality and stability offer complementary signals, while label-driven combinations can outperform the label-free pick on some datasets.

C.5 Selector Studies

  • All 15 reported selection changes lower AUROC by 2.2–44.0 points; Facebook falls to 47.4 from 91.4 in six changes, and Elliptic++ to 30.3 from 72.5 in two.The authors identify the NullKS choice of score type as the fragile step.Source: §C.5 · p. 31, Table 18 · p. 31
  • At ±25% cutoff changes, the selected family and branch remain unchanged in 459 of 462 evaluations (99.4%); at ±50%, they remain unchanged in 450 of 462 (97.4%).Table 18 lists evaluations where a selection changes.Source: Table 18 · p. 31, §C.5 · p. 31
  • Quality alone picks the rule’s score on four datasets, while stability alone never does; the combined rule is within 2.0 points of the best bank member on seven of eight.On Reddit, the rule picks R at 60.6 while the bank’s best member scores 62.9.Source: §C.5 · pp. 31–32, Table 19 · p. 32
  • Label-driven logistic combinations improve on the label-free pick by less than one point on four datasets, 3.1–4.3 on three, and 8.2–17.1 on four.Using those labels leaves the training-free setting; the pools contain more information than the selector extracts, especially on financial graphs and ACM.Source: Table 20 · p. 32, §C.5 · p. 32

C.6 Robustness of the Fit and Scope of the Gaussian Model

This section examines the robustness and scope of EB-GAD’s Gaussian model, including conditions where its scores work or fail and potential deployment harms. It also records the work’s use of human subjects, crowdsourcing, and LLM assistance.

C.6 Robustness of the Fit and Scope of the Gaussian Model

  • Under contamination, Weibo’s original-anomaly AUROC fell from 92.5 to 91.4 as total contamination rose from 4.1% to 24.1%, while the fitted prior shifted from (ρ, κ) = (0.70, 3) to (0.74, 3).On Reddit, the fit stayed at κ = 3 through 13.3% contamination and moved to (0.43, 5) at 23.3%; original anomalies remained at chance. At the anchor bandwidths, fits did not move and original-anomaly AUROC stayed at 95.0–95.2 on Weibo and 60.1–60.6 on Reddit. The likelihood-based fit is expected to degrade gradually, with estimating-equation perturbation of order ϵ times anomalous residual energy.Source: §C.6 · p. 32, Table 21 · p. 33
  • In the Gaussian-model stress test, heavy tails cost at most 0.7 AUROC points, while a two-component normal population cost 9.0–19.3 points.Binarized features yielded selected-score AUROCs of 56–59 (best bank member: 58–61), or 59–61 when anomalies were bit flips (best member: 61–64); the authors identify a categorical residual likelihood as a natural future replacement. Benchmark one-hot features are first passed through PCA, yielding approximately continuous inputs.Source: Table 22 · p. 33, §C.6 · p. 33

C.7 Graph Structure, Edge Uncertainty and Neighboring Anomalies

  • On karate-club connectivity profiles, changing one uncertain edge left the overall ranking stable (rank correlation 0.989), with 98.8% of total absolute score change on the edge endpoints and their neighbors.Structure affects the score through the smoothed template and precision coupling; at γ⋆ = 1, EB fitted ρ = 0.008. At bandwidth 0.7, 74.7% of change remained in the same neighborhood and rank correlation was 0.927.Source: §C.7 · p. 33
  • 92.0–93.6 AUROC for connected injected anomalies exceeded 79.9–83.0 for scattered swaps on Reddit at the EB-fitted bandwidth.At the anchor bandwidth, selected scores were 55.0–57.4 for connected groups and 52.1–77.6 for scattered swaps; the best bank member reached 89.7–91.4 for connected groups. The two bandwidths detect different anomalies: labeled Reddit anomalies are found at the anchor, injected swaps at the fitted bandwidth.Source: Table 23 · p. 34, §C.7 · p. 34
  • For connected injected-node edges, the smaller endpoint energy achieved 94.4–96.7 AUROC, while the standardized cross-term achieved 77.5–84.0.At Reddit’s anchor bandwidth, these readouts fell to 87.5–91.1 and an inverted 8.4–9.6, respectively; large contiguous anomalous regions were not studied and may become their own context.Source: Table 24 · p. 34, §C.7 · p. 34

C.8 Template Variants on BlogCatalog

  • On BlogCatalog, the NullKS-selected equilibrium anchor scored 77.4 AUROC with the paper’s affinity template, versus 75.3 with high-pass and 74.6 with mixed affinity; none approached TAM at 82.5.The best bank scores were 78.6, 76.0, and 74.8, respectively. BlogCatalog has the benchmarks’ lowest feature homophily (h = 0.009) and is routed to the feature-affinity template.Source: §C.8 · p. 34

NeurIPS Paper Checklist

  • The paper states that its abstract and introduction report Table 1 results, including datasets where EB-GAD trails, and clarifies which results depend on the anchor-bandwidth rule.Section 4 and Appendix C.1 describe that rule, adopted after an initial regeneration had been evaluated, and give results without it.Source: NeurIPS Paper Checklist · p. 34
  • The checklist identifies false positives and disparate impact as potential harms of deployment in financial or social settings, and recommends human review, calibrated thresholds, and subgroup-error audits.The paper describes fraud detection, cybersecurity, and scientific networks as target settings.Source: NeurIPS Paper Checklist · p. 38
  • The released code is documented with installation, data sources, protocol, and the script behind each table.Source: NeurIPS Paper Checklist · p. 40
  • The work does not involve crowdsourcing or research with human subjects.The checklist marks the relevant human-subjects and crowdsourcing questions N/A.Source: NeurIPS Paper Checklist · p. 40
  • LLMs are not a component of the method; an assistant was used to polish writing and help orchestrate experiments.Source: NeurIPS Paper Checklist · p. 41
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