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Benchmarking Graph Neural Networks for AC Power Flow: The Role of Inductive Biases and Physical Constraints

Federico Clerici, Ferran Bohigas-Daranas, Hamid Latif-Martínez, Eduardo Prieto-Araujo, Oriol Gomis-Bellmunt, Pere Barlet-Ros

arXiv:2610.38985v1cs.LGcs.AI

TL;DR

AC power flow is a nonlinear task that must be solved frequently, while the effects of GNN architectural choices remain difficult to isolate across differing evaluation settings. This paper benchmarks representative GNNs under a common framework and analyzes errors and design components. Anisotropic, edge-conditioned message passing offers the most favorable accuracy–physical-consistency trade-off in the studied setting.

  • Problem

    Architectural effects on GNN accuracy and physical consistency remain difficult to assess because existing comparisons use differing graph representations and evaluation settings.

    Source: §2.2 · p. 6

  • Method

    The paper benchmarks representative GNN message-passing designs under a unified framework and combines architectural diagnostics with targeted ablations.

    Source: §1 · p. 3, • Phase 3 – Ablation study · p. 4

  • Results

    Anisotropic, edge-conditioned message passing, exemplified by ResGated with two independent layers, offered the most favorable trade-off between statistical accuracy and physical consistency, particularly on IEEE-118.

    Source: §7 · pp. 29–30

  • Takeaways & Limitations

    Within the studied setting, edge features and global context are important components, while residual and root connections were not clearly beneficial at shallow depth.

    Source: §7 · p. 30

    The diagnostic interpretation is grounded in IEEE-118’s weak connectivity and should not be treated as a universal rule for all grids.

    Source: §6 · p. 28

Abstract

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Fast and accurate models for AC power flow (ACPF) are increasingly important as modern electrical grids require more frequent system analyses. Graph neural networks (GNNs) naturally exploit electrical grid topology, yet the effects of architectural design choices on predictive performance remain poorly understood given differences in graph representations and evaluation settings across studies. We present a unified benchmark of representative GNN architectures for ACPF prediction under a common experimental framework. All models use identical preprocessing and training procedures and comparable parameter budgets, isolating the impact of message-passing mechanisms and inductive biases. We also analyze errors by distance from the slack bus and systematically ablate key design components. We evaluate predictive accuracy and physical consistency on the IEEE 14-bus and IEEE 118-bus systems. Results indicate that anisotropic message-passing architectures offer the most favorable trade-off between predictive accuracy and physical consistency under the studied parameter and topology constraints. Explicit edge features encoding line impedances and global context aggregation are particularly important for representing electrical interactions and system-wide dependencies, especially for voltage-angle predictions on the topologically bottlenecked IEEE-118 grid. Under a budget with approximately fixed parameters, two or three message-passing layers generally provide the best balance between depth and representation width, while additional depth offers no consistent improvement. Ablations also show that standard residual and root connections are not necessarily beneficial in shallow models, and that aggregation operators involve trade-offs between statistical accuracy and physical consistency. These results provide practical guidelines for designing graph-based power system models.

1. Introduction

AC power flow is a nonlinear foundation for higher-level grid analyses, and frequent evaluations motivate learning-based approximations. The paper addresses limited understanding of GNN design choices through a controlled benchmark, diagnostics, and ablations.

1. Introduction

  • AC-PF determines steady-state electrical variables and supports higher-level applications such as Optimal Power Flow, while its nonlinear formulation requires iterative numerical solvers.Modern grids require increasingly frequent analyses as renewable generation, electrification, and distributed resources grow.Source: §1 · pp. 2–3
  • GNNs exploit grid topology, but differing graph representations, training settings, parameter budgets, and evaluation protocols make architectural comparisons difficult to interpret.The literature has limited evidence on how message-passing mechanisms and inductive biases affect accuracy, physical consistency, and interactions with power grids.Source: §1 · p. 3
  • The paper evaluates representative message-passing mechanisms under a common heterogeneous graph representation, training protocol, and comparable parameter budget.Its three phases comprise a broad benchmark, diagnostics of top-performing architectures, and ablations of structural components.Source: §1 · p. 3, • Phase 3 – Ablation study · p. 4

2.2. Existing GNN Benchmarks and the Case for Architectural Analysis

  • Prior benchmarks standardize datasets or evaluation but often cover few architectures and omit targeted ablations linking performance to physical properties such as KCL satisfaction and impedance sensitivity.Other comparisons leave component contributions unclear or do not control parameter budgets.Source: §2.2 · pp. 6–7
  • The analysis examines error by topological distance from the slack bus and uses ablations to isolate how inductive biases relate to physical constraint satisfaction.The stated scope is fixed grid topologies with varying power injections, rather than dynamic topology shifts such as N −1 contingencies.Source: • Phase 3 – Ablation study · p. 4, §2.2 · p. 7

3. Methodology

The paper uses the message-passing framework to describe how GNNs exchange information across a graph. This common representation makes architectural differences interpretable as differences in message generation, aggregation, and node updating.

3.1. The Message-Passing Framework

  • At each layer, a message function computes interactions between a target node and its neighbors, optionally using edge features, and a permutation-invariant operator pools the messages.The resulting neighborhood message is passed to an update function that combines it with the node’s previous state.Source: §3.1 · p. 8

3. Methodology

  • The benchmark treats diverse GNN architectures as instantiations of message generation, aggregation, and node-update operations, enabling comparison of their relational inductive biases.This abstraction is intended to make computational mechanics interpretable for physical network analysis.Source: §3 · p. 7, §3.1 · p. 8

3.2. Message Passing Taxonomy

  • The evaluated architectures are classified by convolution domain, treatment of neighbor importance, and integration of edge attributes.These axes organize the benchmark’s comparison of message-passing designs.Source: §3.2 · p. 8

Domain of Operation

The architecture taxonomy distinguishes where convolutions operate, how neighbors are weighted, and how edge attributes enter message passing. The benchmark also standardizes model components to focus comparisons on message-passing formulations.

Domain of Operation

  • Spatial models pass messages along graph edges, whereas spectral models use Laplacian eigendecomposition and are excluded because their filters depend on graph eigenstructure.The stated concern is that spectral filters favor topology-specific patterns rather than local information-flow rules.Source: Domain of Operation · pp. 8–9

Treatment of Neighbor Importance

  • Isotropic models apply the same transformation across neighbors, while anisotropic models use feature-dependent filters to assign differing importance to edges.GCN and GIN are examples of isotropic models; GAT and ResGated exemplify anisotropic models.Source: Treatment of Neighbor Importance · p. 9

Integration of Edge Attributes

  • Topology-only models use edges to define neighborhoods, while edge-conditioned models incorporate continuous edge features into message computation.The latter transforms messages using the connecting edge’s feature vector, which suits interactions governed by connection parameters.Source: Integration of Edge Attributes · pp. 9–10

3.3. Detailed Architecture Specifications Shared vs. Independent Weights

  • The paper compares independently parameterized layers with repeated application of a shared transformation, interpreting shared weights as a recurrent refinement bias.The comparison is motivated by the iterative structure of classical AC-PF solvers.Source: Shared vs. Independent Weights · p. 10

Modeling Self-Information in Heterogeneous Power Grids

  • Architectures that need an explicit self-contribution use a root-weight transformation rather than changing the heterogeneous graph topology.GCN, GAT, and Graph Transformer use this approach; other evaluated architectures preserve self-information through their native formulations.Source: Modeling Self-Information in… · pp. 10–11

3.4. Heterogeneous Message Passing Pipeline

  • All architectures use a standardized three-stage heterogeneous pipeline of type-specific encoding, message-passing backbone, and node-specific decoding.This design is intended to isolate performance differences to the evaluated message-passing formulations.Source: §3.4 · p. 11, Figure 1 · p. 11

Input Encoding

The study encodes heterogeneous grid inputs, applies message passing, and decodes bus states within a controlled benchmark. It evaluates two fixed IEEE topologies using generated load variations and consistent training and evaluation conditions.

Input Encoding

  • Each bus type receives a dedicated linear projection to a shared hidden dimension followed by Layer Normalization, while directed-edge impedance features are projected and reused across layers.The encoded edge vector includes resistance and reactance and uses a SiLU activation.Source: Input Encoding · p. 12

3.5. Message-Passing Backbone

  • The backbone compares distinct layer parameters with a shared layer applied repeatedly alongside a per-type GRU update.The shared-weight configuration is described as unrolling an iterative physical AC-PF solver.Source: §3.5 · p. 12
  • Node updates combine transformed incoming messages with residual and root connections, and may broadcast a mean-pooled global context vector to every node.The residual connection is described as a way to mitigate oversmoothing; global context supplies a soft global signal.Source: Residual, Root, and Global Connections · p. 12

Output Decoding

  • Separate two-layer regression heads predict voltage magnitude and angle for each node type, while the slack bus has no decoder because its states are fixed setpoints.Separate heads accommodate the targets’ distinct physical scales and distributional asymmetries.Source: Output Decoding · p. 12

Architectural Harmonization

  • The benchmark harmonizes edge-feature ingestion, fixes sum pooling for Phase 1, and uses topology-neutral root transformations where explicit self-loops would otherwise be required.For attention models, edge features enter attention scores; Phase 3 tests aggregation choices for the winning architecture.Source: Architectural Harmonization · p. 13

4. Experimental Setup

  • The experimental setup defines grid topologies, data generation, graph representation, training, and evaluation under consistent conditions.IEEE-118 has a substantially larger diameter and average shortest-path length and nearly 20 times smaller algebraic connectivity than IEEE-14, indicating a stronger bottleneck for long-range propagation.Source: §4 · p. 13, §4.1 · pp. 13–14, Table 3 · p. 14

4.2. Dataset Generation

  • For each IEEE topology, the study generates 2,000 feasible states with active and reactive demands perturbed by ±80% and line impedances by ±5%, discarding non-convergent samples.The load range is described as broad and wider than in several previous studies.Source: §4.2 · p. 14

Graph Representation

The paper represents each grid state as a heterogeneous graph whose bus types, input features, line attributes, and voltage targets reflect the AC power-flow setup. It then standardizes the data, trains the models, and evaluates both statistical accuracy and physical feasibility.

Graph Representation

  • PV, PQ, and slack buses are distinct node types; inputs encode their known conditions, while targets are voltage magnitudes at PQ buses and angles at PQ and PV buses.Edges are bidirectional and carry line resistance and reactance; predicted voltage magnitude and angle are sufficient to define the network state.Source: Graph Representation · pp. 14–15, Table 4 · p. 15

4.3.1. Data Splitting and Standardization

  • Samples use a 75%/10%/15% train-validation-test split across four fixed seeds, and Z-score statistics are fit on training data to prevent information leakage.Standardization is performed independently for each node type and feature.Source: §4.3.1 · p. 15

4.3.2. Training Configuration

  • Training minimizes separate MSE losses for voltage magnitude and angle, using Adam and early stopping across four seeds.Independent target standardization makes the two loss components comparably weighted despite different physical scales.Source: §4.3.2 · p. 16

4.4. Evaluation Metrics

Thresholded Relative Mean Absolute Error (TRMAE)

  • TRMAE reports relative percentage error separately for voltage magnitude and angle, and by bus type; τ = 0.001 limits instability when targets approach zero.The threshold prevents undefined or artificially inflated relative errors near zero.Source: Thresholded Relative Mean Absolute… · p. 16

Mean Absolute Power Imbalance (KCL-MAE)

The paper supplements statistical accuracy with KCL-MAE, a physical-consistency metric based on complex power imbalance. Its benchmark compares architectures under a shared parameter budget, then diagnoses errors and ablates selected design components.

Mean Absolute Power Imbalance (KCL-MAE)

  • KCL-MAE measures complex power imbalance at each bus from predicted voltages and scheduled injections; lower values indicate closer alignment with Kirchhoff’s conservation laws, not just statistical fit.Active and reactive mismatch components separately identify violations of active and reactive power balance.Source: Mean Absolute Power Imbalance (KCL-MAE) · pp. 16–17

4.5. Benchmark Structure

  • Configurations are constrained to approximately 100k trainable parameters, with hidden dimensions adjusted across message-passing depths and propagation settings.The budget is intended to distinguish architectural effects while keeping experiments computationally feasible.Source: §4.5 · pp. 17–18

4.5.1. Phase 1 – Broad Architecture Benchmark

  • Phase 1 compares seven GNN architectures and an MLP baseline using TRMAE and KCL-MAE under common representations, training protocols, and comparable parameter budgets.This setup is designed to make differences primarily reflect message-passing mechanisms rather than model capacity or optimization settings.Source: §4.5.1 · p. 18

4.5.2. Phase 2 – Architectural Diagnostics

  • Phase 2 analyzes prediction error by shortest-path distance from the slack bus to examine how longer information-propagation distances relate to performance.A complementary spectral error analysis is reported in the appendix.Source: §4.5.2 · p. 18

4.5.3. Phase 3 – Ablation Study

  • Phase 3 removes or modifies one component at a time in targeted ablations, selecting five components for their theoretical and physical grounding.The ablation targets and their guiding hypotheses are summarized in Table 5.Source: §4.5.3 · p. 18, Table 5 · p. 19

5. Results

Across the studied settings, independent ResGated layers provide the strongest physical-consistency result, while statistical accuracy rankings vary across targets and bus types. Diagnostics and ablations on IEEE-118 link prediction errors to distance from the slack bus and identify consequential architectural choices.

Statistical Accuracy

  • 16.5% for voltage angles and 15.9% for voltage magnitudes are the average error increases from shared versus independent weights; T = 2 or T = 3 generally performs best under the fixed budget.T = 4 often has higher variability, and the depth pattern reflects capacity allocation between width and depth rather than a general penalty for deeper propagation.Source: Statistical Accuracy · pp. 19–20

Physical Consistency

  • Independent ResGated (T = 2) achieves the lowest active power mismatch at 0.4964 p.u. and is selected for subsequent diagnostics and ablations.Statistical accuracy alone does not yield a clear winner across predicted quantities and bus types.Source: Statistical Accuracy · p. 20, Physical Consistency · p. 22
  • Independent layers outperform shared weights, while anisotropic ResGated gates messages using branch parameters and has lower empirical KCL residuals than isotropic counterparts.The study attributes this contrast to the gate’s multiplicative scaling of messages and its prioritization of paths based on line admittances.Source: Physical Consistency · p. 22

Distance from the Slack Bus Analysis

5.3. Phase 3: Ablation Study

  • Omitting edge features raises PQ voltage-angle error to 24.17%, while removing global pooling increases active power mismatch by nearly 87%, from 0.4964 to 0.9292 p.u.The authors connect these degradations to missing line-impedance information and loss of global context, respectively; Table 6 reports ablation results.Source: §5.3 · p. 26, Table 6 · p. 27
  • Removing residual connections yields ∆P = 0.3148 p.u. and ∆Q = 0.0381 p.u.; removing root connections also improves KCL-MAE in the shallow T = 2 model.The authors suggest identity bias from these connections can add unpropagated initial-state noise in shallow models.Source: §5.3 · p. 26, Table 6 · p. 27
  • Replacing baseline sum aggregation with mean or max reduces voltage-angle prediction errors: PQ is 9.78% with mean, and PV is 7.41% with max; only max significantly improves KCL-MAE for ΔP.Mean’s KCL-MAE effect is small and non-robust (dz = 0.50; bootstrap intervals include zero), while max has dz = 1.19 for ΔP.Source: §5.3 · p. 26
  • The ablations recommend retaining edge attributes and global pooling, while considering removal of residual and root connections and use of max aggregation for shallow models.The authors frame these as design rules for the studied graph-based AC-PF modeling setting.Source: §5.3 · p. 27

6. Discussion

The benchmark does not identify a universally dominant architecture; performance depends on how capacity is allocated across width, depth, edge processing, and global context. Its findings favor shallow-to-moderately-deep, edge-conditioned models with global context, within the study’s tested scope.

6. Discussion

  • Performance depends on capacity allocation across representation width, propagation depth, local edge processing, and global context rather than on a universally dominant architecture.Source: §6 · p. 27
  • Two- or three-layer configurations tend to balance depth and width favorably, while additional message-passing rounds may be less efficient than explicit global aggregation for system-wide dependencies.Because the parameter budget is fixed, increasing depth shrinks hidden dimensions; the results do not isolate depth’s causal effect.Source: §6 · pp. 27–28
  • Independent-layer configurations generally outperform shared recurrent counterparts, especially at two and three layers, but the comparison also changes the update mechanism by introducing a GRU.Source: §6 · p. 28
  • Voltage-angle error degrades with distance from the slack bus and depends more strongly on global pooling, while voltage-magnitude error remains small and largely distance-insensitive.The proposed link between topological bottlenecks and benefits from global context remains a hypothesis based on two grids.Source: §6 · p. 28
  • Removing edge features degrades both statistical accuracy and physical consistency, and comparable TRMAE can coexist with differing KCL-MAE, making physics-aware metrics relevant to model selection.Source: §6 · p. 28
  • Aggregation and connection choices involve trade-offs: mean or max aggregation and removing residual or root connections improve several metrics in the tested shallow, two-layer model.Mean aggregation is stable for continuous nodal quantities, while max-pooling may emphasize dominant local power flows for KCL.Source: §6 · pp. 28–29
  • The conclusions are bounded by two fixed IEEE topologies, in-distribution load perturbations, a fixed parameter budget that entangles depth and width, post-hoc physical residuals, and four seeds.Source: §6 · p. 29

7. Conclusion

Under the controlled benchmark, anisotropic edge-conditioned message passing—exemplified by ResGated with two independent layers—offered the most favorable accuracy–physical-consistency trade-off. The resulting design guidance is a starting point limited by the studied topologies, load perturbations, parameter budget, and post-hoc assessment of physical constraints.

7. Conclusion

  • The shared parameter budget prevents fully disentangling propagation depth from representation width, so results compare capacity-allocation strategies rather than unconstrained architecture maxima.The authors also note that more random seeds would improve statistical robustness and runtime and memory measurements would support deployment-oriented selection.Source: §7 · pp. 29, 31
  • ResGated with two independent layers offered the most favorable trade-off between statistical accuracy and physical consistency, particularly on IEEE-118.This qualitative pattern appeared across the benchmark, diagnostics, and ablations.Source: §7 · pp. 29–30
  • For weakly connected grids such as IEEE-118, favor anisotropic, edge-conditioned message passing when physical consistency is a priority under the study’s protocol.The authors identify edge features and global context aggregation as components the model could least afford to lose.Source: §7 · p. 30
  • Treat global context aggregation as necessary rather than optional, especially for voltage-angle prediction and topologically weakly connected grids.Residual and root connections were not clearly beneficial at the tested shallow depth.Source: §7 · p. 30
  • Evaluate residual and root connections with physical-consistency metrics rather than assuming they help based on results in other domains.The benchmark also cautions that statistical and physical metrics can disagree.Source: §7 · p. 30
  • These guidelines are starting points, not fixed rules, and were derived on two fixed IEEE topologies with in-distribution load perturbations and post-hoc physical residuals.The study calls for testing topology changes, N −1 contingencies, larger and differently structured grids, and physics-enforcing training approaches.Source: §7 · p. 30

Appendix A. Extended Experimental Setup and Implementation

The appendices document parameter-controlled architecture configurations, implementation choices, and the complete Phase 1 result materials. The provided passages specify the benchmark’s approximate parameter budget and the scope of its result tables and plots, but do not report their cell values.

Appendix A.1. Model Architectures and Hyperparameters

  • Configurations scale hidden dimensions to keep evaluated architectures near a 100,000-parameter budget across independent and shared weight modes.Table A.7 specifies message-passing depth, hidden dimensions, and total parameter counts.Source: Table A.7 · p. 36

Appendix A.2. Additional Architecture Implementation Specifications

  • Implementation variants adapt topology handling, edge-feature injection, aggregation, and self-identity mechanisms to each architecture.Examples include GraphSAGE’s sum aggregation in Phase 1, GINE’s intrinsic self-identity term, ResGated’s dimension-wise edge gates, and topology-restricted Transformer attention.Source: Appendix A.2 · pp. 36–37

Appendix B.1. Phase 1: Complete Results

  • Phase 1 reports complete statistical and physical performance results through aggregated tables and comparative plots.The supplied captions identify TRMAE by bus type and KCL mismatch across datasets, modes, and message-passing depth; they do not provide result values.Source: Appendix B.1 · p. 37, Table B.8 · p. 38, Table B.9 · p. 39

Appendix B.2. Phase 1: Correlation Between Error (TRMAE) and KCL

The appendix compares statistical prediction error with physical mismatch and uses KCL-MAE as the decisive model-selection metric because the two measures do not consistently align. ResGated has the lowest reported mean mismatch point estimates, though the limited seed count constrains statistical testing.

Appendix B.2. Phase 1: Correlation Between Error (TRMAE) and KCL

  • ResGated (T = 2) has the lowest mean mismatch point estimates for both active and reactive power, but several top models’ error bars overlap.With n = 4 seeds, standard hypothesis testing is severely underpowered and relies on unverifiable normality assumptions.Source: Figure B.16 · p. 44, Appendix B.2 · p. 45, Figure B.17 · p. 45
  • Lower TRMAE does not perfectly correlate with lower physical mismatches, particularly on IEEE-118, so KCL-MAE is designated decisive for model selection.The figures compare active and reactive power mismatch with average TRMAE across bus types.Source: Figure B.16 · p. 44, Appendix B.2 · p. 45, Figure B.17 · p. 45
  • Active-power mismatch effect sizes favor ResGated over GAT (dz = −0.94) and Transformer (dz = −0.75), with confidence intervals excluding zero.For reactive-power mismatch and comparisons against GraphSAGE, effects are small-to-medium with intervals containing zero; ResGated is selected for subsequent diagnostics and ablations.Source: Appendix B.2 · p. 46, Table B.10 · p. 46

Appendix B.3. Phase 2 diagnostic: Distance from the Slack Bus Analysis – Other Models

The appendix supplies additional slack-bus-distance plots for five architectures and frames spectral decomposition as a complementary diagnostic of localized versus global behavior. Its spectral analysis uses an admittance-weighted structural proxy rather than the full complex electrical coupling.

Appendix B.3. Phase 2 diagnostic: Distance from the Slack Bus Analysis – Other Models

  • The GCN, GAT, GINE, GraphSAGE, and Transformer plots report IEEE-118 performance by hop distance from the slack bus for T = 2 independent layers, with mean and variance across seeds.The supplied figure captions describe the comparison but do not state its outcomes.Source: Figure B.18 · p. 47, Figure B.19 · p. 47, Figure B.20 · p. 48

Appendix B.4.1. Theoretical Framework: Spectral Error Decomposition

  • The spectral diagnostic decomposes nodewise residuals in a graph-spectral basis to examine localized bus interactions and global grid dynamics beyond point-wise error.The text motivates comparing spectral error profiles across isotropic aggregators and anisotropic gated or attention-based architectures.Source: Appendix B.4.1 · pp. 49–50

Defining the Admittance-Weighted Laplacian Proxy

  • The proxy weights edges by admittance magnitude, encoding coupling strength while discarding complex phase and the distinct roles of conductance and susceptance.It supports a real-valued structural basis for this diagnostic and is described as closer to grid physics than binary topological adjacency.Source: Defining the Admittance-Weighted… · p. 50
  • The symmetric normalized Laplacian’s eigenvalues act as graph frequencies: low values represent smooth variation across strongly coupled buses, while high values represent rapid edge-to-edge oscillations.The Laplacian is eigendecomposed as L = UΛU⊤.Source: Defining the Admittance-Weighted… · p. 50

Residual Projection and Cumulative Energy

On IEEE-118, spectral diagnostics show that voltage-angle errors concentrate in low-frequency modes and that removing global pooling sharply worsens angle residual energy. The interpretation is limited by the spectral proxy and unresolved topology-versus-capacity effects.

Residual Projection and Cumulative Energy

  • The analysis projects nodewise prediction residuals into graph-frequency modes and examines mean and cumulative spectral energy to characterize error distribution.More rapidly increasing cumulative energy indicates smoother, low-frequency error; slower growth indicates more localized, high-frequency mismatches.Source: Residual Projection and Cumulative… · pp. 50–51

Assumptions and Limitations

  • The normalized-Laplacian proxy discards complex electrical-coupling phase, so its eigenvectors are diagnostic modes rather than exact physical wave-propagation modes.The analysis assumes a fixed topology over the evaluated steady-state benchmark.Source: Assumptions and Limitations · p. 51

Appendix B.4.2. Diagnostic Findings: Network Anchoring and Topological

  • Figure B.23 shows baseline Va errors accumulating in the lowest eigenmodes, while Vm errors extend more gradually into higher-frequency modes.The pattern is consistent with global angle shifts and more localized voltage regulation.Source: Appendix B.4.2 · p. 51, Figure B.23 · p. 52
  • Va’s low-frequency residuals can reflect a uniform DC offset: preserving smooth relative bus differences does not ensure angles are anchored to the slack-bus reference.The discussion associates this limitation with isotropic aggregators’ low-pass responses and contrasts them with the greater flexibility of anisotropic, gated ResGated.Source: Appendix B.4.2 · p. 52
  • Removing global pooling broadens Va’s cumulative error distribution into a step-like shape, while Vm’s distribution changes little.The proposed slack-bus and topological-cluster explanations are suggestive, not independently identified.Source: Appendix B.4.2 · pp. 52–53
  • IEEE-118 eigenvalue clusters are consistent with topological bottlenecks, but the spectral-gap argument alone cannot distinguish oversquashing from capacity or depth effects.Separating these effects would require relating spectral gaps to depth-wise error curves while decoupling message-passing depth from the parameter budget.Source: Figure B.24 · p. 53, Appendix B.4.2 · pp. 54–55
  • Figure B.25 shows that removing global pooling sharply degrades Va residual energy, especially in low- and mid-frequency bands, and roughly quadruples Vm error in those bands.Vm residual energies remain below 4 × 10^-5; the analysis uses fixed spectral-band boundaries.Source: Appendix B.4.2 · pp. 53–54, Figure B.25 · p. 54

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