Batch Before You Lift: Scalable Topological Deep Learning on Large Graphs

1 INTRODUCTION

• Higher-order domains represent multi-way interactions that pairwise graphs cannot explicitly represent, providing complementary inductive biases for such systems.Examples include coauthorship groups, molecular rings, and many-body couplings.Source: §1 · p. 1
• Reddit's approximately 233k nodes and 57.3M edges yield approximately 8.36B simplicial 2-cells under clique-complex lifting, making full-domain construction a memory bottleneck.The lifted domain can be orders of magnitude larger than the input graph because its size depends on connectivity, not node count alone.Source: §1 · p. 2
• Existing mini-batching often begins after lifting, so it still requires materializing the complete higher-order domain and may use domain-specific procedures.Batching before lifting instead operates on the graph shared across higher-order constructions, but independently lifted partitions would omit cross-boundary edges and structures.Source: §1 · p. 2
• Cluster-TNN samples groups of graph clusters, reconstructs their induced subgraph with all original edges among sampled nodes, then lifts that mini-batch for a TNN.Reconstruction restores edges between jointly sampled clusters, allowing structures to be recovered when all their supporting nodes occur in one batch.Source: §1 · pp. 2–3
• Across 21 matched model-dataset comparisons, Cluster-TNN reduces peak GPU memory by 83.2% on average while maintaining competitive predictive performance.The framework reuses existing liftings and TNN architectures, and enables evaluation on Reddit and OGBN Products.Source: §1 · p. 3

2 RELATED WORK

• TopoX supplies topological data structures and neural operators, while TopoBench integrates datasets, liftings, and models for comparisons across higher-order architectures.TopoBench reports out-of-memory errors during clique and cycle lifting on graphs with fewer than 25,000 nodes.Source: §2 · p. 3
• Existing scalability methods subsample constructed hypergraphs, accelerate inference, batch neighborhoods using node-to-cell incidence, or precompute features and encodings.These methods operate on higher-order domains that have already been constructed, leaving domain construction across lifting families as a complementary objective.Source: §2 · p. 3
• Graph mini-batching methods include neighborhood sampling, layer-wise importance sampling, and subgraph sampling; Cluster-GCN groups node clusters while retaining edges among selected nodes.For graph-supported TDL, retaining those edges matters because missing edges can prevent higher-order structures from being constructed even when all supporting nodes are present.Source: §2 · p. 3

3 METHODS

• Cluster-TNN moves mini-batching before higher-order domain construction, controlling both domain construction and model execution without materializing the full lifted domain.It reconstructs induced subgraphs from grouped clusters, retains edges among sampled nodes, and applies the selected lifting before TNN execution.Source: §3 · p. 4

3.1 MINI-BATCH CONSTRUCTION AND LIFTING

• At each epoch, clusters containing training nodes are grouped into batches of at most q clusters, and each batch's induced subgraph retains all edges among its nodes.The selected lifting constructs the higher-order elements and features; the TNN uses the same interface as in full-domain execution.Source: §3.1 · p. 4
• The workflow requires a lifting evaluable on an induced graph and a TNN that processes variable-size lifted domains through the full-domain interface.Architectures requiring operators or state on the complete lifted domain need adaptation; native higher-order datasets require batching in that domain instead.Source: §3.1 · p. 4

3.2 TRAINING AND INFERENCE

• Each lifted mini-batch produces one optimization step using cross-entropy loss on its supervised training nodes.The loader uses active training clusters, so each mini-batch contains supervised nodes regardless of their number.Source: §3.2 · p. 4
• Inference can use one cluster grouping, average predictions over independent groupings, or materialize the full lifted graph when memory permits.The primary partitioned results use ensemble mini-batch inference with ten independent cluster recombinations.Source: §3.2 · pp. 4–5

3.3 THEORETICAL GUARANTEES

• A structure is recoverable only when its supporting clusters fit in one batch: structures spanning at most q clusters can be recovered under lifting compatibility.Structures spanning more than q clusters cannot appear in a q-cluster batch.Source: §3.3 · p. 5
• With independent cluster recombinations, the probability of observing a structure after T epochs is 1 − (1 − p_s)^T when its cluster span is at most q; if q is at least the maximum span, cumulative recovery approaches the full structure set.This guarantee assumes a fixed partition, uniform independent permutations of all K clusters, q dividing K, and lifting compatibility.Source: §3.3 · p. 5
• At q = 4, recovery exceeds 99% for cellular and simplicial 2-cells on Cora Full; at q = 100, Coauthor Physics reaches 59.3% and 58.9%, respectively.These are fractions of reference structures whose supporting nodes are jointly sampled within 200 epochs; larger supports can span more clusters and reduce joint-sampling probability.Source: Figure 2 · p. 5, §3.3 · pp. 5–6

4.1 EXPERIMENTAL SETUP

• Evaluation spans six benchmarks across controlled, intermediate-density, and large-scale regimes, comparing GCN with hypergraph, cellular, and simplicial models.Full-graph comparisons are confined to the controlled regime because at least one higher-order model exceeds available resources in the other regimes; OGBN Products uses generated random splits rather than its official sales-ranked split.Source: §4.1 · pp. 6–7

4.2 COMPUTATIONAL EFFICIENCY

• 83.2% mean peak-GPU-memory reduction across 21 matched comparisons, with reductions in every comparison.The mean reduction is 83.2% and the median is 86.7%.Source: Figure 3 · p. 7, §4.2 · p. 7
• At depth 16, Cluster-TNN peaks remain below 20 GB for TopoTune and SCCNN, versus approximately 69–80 GB for full-graph execution.On Reddit, full-graph TopoTune preprocessing takes 6,965.2 s and reaches 480.1 GB peak CPU memory, while Cluster-TNN preprocessing takes 500.4 s and run-wide peak CPU memory is 11.81 GB.Source: Figure 4 · p. 7, §4.2 · pp. 7–8

4.3 FEASIBILITY AND PREDICTIVE PERFORMANCE

• Cluster-TNN reports a higher mean test metric in 13 of 21 matched comparisons; on OGBN Products, SCCNN reaches 88.63% accuracy versus 85.81% for GCN.On Coauthor Physics, higher-order models reach 94.97–96.31% mean accuracy versus 96.68% for GCN; on Reddit, higher-order TNN accuracies are comparable to the GCN baseline.Source: Table 1 · p. 8, §4.3 · p. 8

4.2 COMPUTATIONAL EFFICIENCY

• Preprocessing time decreases in 16 of 21 comparisons, while total runtime decreases in 11 and increases in 10.Among preprocessing reductions, the mean is 53.9% and median 53.1%; total-runtime reductions have a median of 55.6%, while increases have a median of 136.9%. Final evaluation can outweigh training savings, as for CWN on Cora Full.Source: §4.2 · p. 8

4.4 ABLATION STUDIES

• METIS yields the higher mean test metric in 15 of 21 comparisons, including six of seven models on Cora Full and Amazon Ratings, but only three on Questions.Its predictive advantage depends on the model and dataset.Source: §4.4 · p. 9
• Preserving cross-cluster edges yields the higher mean test metric in 19 of 21 comparisons, with a 10.01-percentage-point maximum difference for TopoTune on Cora Full.These ablations support retaining the full induced connectivity among co-sampled clusters.Source: §4.4 · p. 9
• Ensemble mini-batch inference beats a single pass in all 21 comparisons, with gains from 0.04 to 4.65 percentage points.These gains require additional inference passes, whose computational cost is included in the runtime comparison.Source: §4.4 · p. 9

5 CONCLUSIONS AND DISCUSSION

• Across 21 matched full-graph comparisons, Cluster-TNN reduces peak GPU memory by 83.2% on average while retaining competitive predictive performance and enabling six higher-order models on Reddit and OGBN Products.The discussion also reports higher mean test metrics for ensemble than single-pass inference in all 21 controlled comparisons.Source: §5 · p. 9
• Structural recovery does not establish equivalence to full-graph predictions or gradients, its role in ensemble gains remains unresolved, and repeated lifting and ensemble inference can increase runtime.The evaluation covers selected liftings for graph-supported node classification but does not establish which representation best suits each dataset and model.Source: §5 · p. 9
• Future work includes batching native higher-order cells and incidences, evaluating selective-caching trade-offs, and testing whether broader context justifies its costs under fixed memory and runtime budgets.Changing model rankings across datasets motivate studying lifting and batching jointly.Source: §5 · p. 9

REPRODUCIBILITY STATEMENT

• The reproducibility statement identifies the computing hardware and provides a code-implementation link.Experiments used eight NVIDIA A100 GPUs, each with 80 GiB of GPU memory.Source: REPRODUCIBILITY STATEMENT · p. 15

AI USE STATEMENT

• The authors report using AI assistance for manuscript and proof exposition tasks, with all AI-assisted content reviewed, revised, and verified by the authors.They state that AI tools did not substitute for scientific judgment, experimental analysis, or verification of results.Source: AI USE STATEMENT · p. 15

ETHICS STATEMENT

• The ethics statement identifies no broader societal implications warranting specific discussion.Source: ETHICS STATEMENT · p. 15

A FEATURED TOPOLOGICAL DOMAINS AND LIFTING MAPS

• Topological domains provide a shared representation language for graphs and higher-order relations, including hypergraph set-relations, simplicial face hierarchies, and cell-complex boundary relations.The appendix also describes the experimental liftings as using neighborhoods, cliques, and graph cycles to instantiate these domains.Source: §A.1 · p. 16

A.2 LIFTING MAPS

• A lifting separates structural construction of target elements and incidences from feature transfer or construction; in this study, featured graphs map to model-required domains.The notation guide collects symbols used across appendix sections, while symbols introduced only in proofs remain local to those proofs.Source: NOTATION GUIDE · p. 15, §A.2 · p. 16

A.2.1 LIFTING MAPS USED IN THE EXPERIMENTS

• The evaluated liftings include identity graphs, k-hop neighborhood hypergraphs, cycle-basis cell complexes, and truncated clique complexes, with construction choices specified for each model pairing.The experiments use k = 1 neighborhoods, cycles of length at most nine, and clique complexes through dimension two.Source: §A.2.1 · p. 17
• Higher-rank feature vectors are formed by summing features over incident lower-rank elements, without changing the structure produced by the structural lifting.For hypergraphs, the analogous assignment sums incident node features for each hyperedge.Source: §A.2.1 · p. 18

B.1 PIPELINE VIEW OF GRAPH-SUPPORTED TDL SCALABILITY

• Full-domain execution constructs and reuses the complete lift, whereas Cluster-TNN repeatedly reconstructs and lifts induced mini-batches, trading full-domain storage for local construction and changed runtime.Applying the same lifting procedure to induced subgraphs need not produce the restriction of the full-domain lift.Source: §B.1 · p. 18

B.2 REPRESENTATION GROWTH: FROM STRUCTURE TO STORAGE

• Lifted-domain storage depends on generated element counts, incidence or neighborhood-operator entries, and feature widths; models may request different operators from the same domain.Source: §B.1 · p. 18

B.2.1 LIFTING-SPECIFIC MECHANISMS OF STRUCTURAL EXPANSION

• Neighborhood lifting has exactly |V| indexed hyperedges but |V| + 2|E| memberships, while cycle and clique lifts add 2-cells according to retained cycle lengths and graph triangles, respectively.Each clique-derived 2-cell contributes one feature row and three edge-cell incidence entries.Source: §B.2.1 · p. 19

B.2.2 BENCHMARK SCALE REGIMES AND COVERAGE

• Reddit has 174.20 cellular and 35886.67 simplicial 2-cells per node, compared with 13.98 and 253.60, respectively, for OGBN Products.The two graphs have comparable edge counts, while OGBN Products has approximately 10.5× more nodes.Source: §B.2.2 · pp. 19–20
• The benchmarks combine small controlled comparisons, an intermediate Coauthor Physics case, and Reddit and OGBN Products with tens of millions of edges and contrasting node counts.They also span online interaction, scientific-information, and product co-purchasing networks while keeping node classification as the downstream task.Source: §B.2.2 · pp. 19–20

B.2.3 STATIC FEATURE-MATRIX STORAGE

• The storage estimates cover persistent input feature matrices only, not total preprocessing or training memory.They assume float32 values and feature widths matching each dataset's input width for generated elements.Source: §B.2.3 · p. 20
• For OGBN Products, estimated float32 feature-matrix storage is 39.414 GB for cellular and 274.156 GB for simplicial liftings.The estimates use 100-dimensional features and the structural counts reported for that dataset.Source: §B.2.3 · p. 20
• For Reddit, estimated simplicial feature-matrix storage is 20.270 TB, before adding incidence tensors or neural-network state.The estimate uses a 602-dimensional float32 feature vector for each node, edge, and generated higher-order element.Source: §B.2.3 · pp. 20–21

B.3 FULL-DOMAIN CPU PREPROCESSING: CONSTRUCTION TIME AND MEMORY

• The preprocessing measurements report available dataset-lifting-model configurations because models may request different operators from the same lifted domain.Peak CPU memory is the largest sampled resident-set size for the main Python process; sampling may miss short-lived peaks and excludes memory held exclusively by child processes.Source: Table 5 · p. 21, §B.3 · p. 21
• For affected OGBN Products configurations, full-graph feature matrices exceed available CPU RAM, and full-graph training is expected to exceed available GPU memory, so partitioned results are reported instead.Source: §B.3 · p. 22
• 4684.8 ± 304.8 s and 479.96 ± 0.12 GB: Reddit CWN full-domain preprocessing time and peak CPU memory across five runs.The cellular feature matrices alone are estimated at 236.280 GB, before incidence matrices, model-requested operators, and temporary allocations.Source: §B.3 · p. 22
• For Reddit, estimated full-graph simplicial input feature matrices require 20.270 TB at the original feature width in float32.Source: §B.3 · p. 22

B.4 PHASE-SPECIFIC MEMORY DURING PREPROCESSING AND TNN EXECUTION

• TNN layers may retain intermediate messages for backpropagation, adding GPU memory beyond input features and connectivity operators.For example, an edge representation can receive separate messages from incident nodes and 2-cells.Source: §B.4 · p. 23
• Cluster-TNN lowers peak GPU memory during training and evaluation, while preprocessing memory varies by configuration and can increase when partition construction costs more than full-graph lifting.Figure 5 reports means across five seeds.Source: §B.4 · p. 23, Figure 5 · p. 23

B.5 END-TO-END COST

• End-to-end scalability measures elapsed time across workflow phases and peak CPU/GPU memory, since reduced execution memory can accompany increased lifting, validation, or inference time.Figure 5 places preprocessing CPU-memory peaks alongside training and evaluation GPU-memory peaks for the seven evaluated models.Source: §B.4 · p. 23, §B.5 · p. 24

B.5.1 END-TO-END RUNTIME

• Mean total runtime is lower for all matched GCN and hypergraph configurations, higher for all cellular configurations, and lower for simplicial configurations only on Questions.The authors describe this as a pattern in evaluated configurations, not an intrinsic runtime ordering of domains.Source: §B.5.1 · pp. 24–25, Table 6 · p. 24
• Training time was lower in 12 comparisons by 60.2% on average (median 63.3%) and higher in nine by 203.9% on average (median 138.5%).Partitioned preprocessing was shorter in 16 comparisons, longer in four, and unchanged in one.Source: §B.5.1 · p. 25
• The pooled runtime aggregate weights each configuration’s relative change by its full-graph runtime, and slower configurations’ additional runtime outweighs faster configurations’ savings across the matched suite.Partitioned-only configurations are excluded from the pooled total-runtime changes.Source: §B.5.1 · p. 25, Table 7 · p. 25
• For CWN on Cora Full, training time fell from 909.6 to 776.2 seconds, but total runtime rose 12.4% as final evaluation increased from 14.6 to 272.4 seconds.The additional evaluation cost outweighs the training-time reduction in this configuration.Source: §B.5.1 · pp. 25–26
• Final evaluation was longer in all 21 matched comparisons, with a median partitioned-to-full-graph time ratio of 11.75×.It includes checkpoint validation and test inference; the measured difference includes repeated evaluation and batch-construction costs and is not an intrinsic lower bound for partitioned inference.Source: §B.5.1 · pp. 24, 26, Table 6 · p. 24

B.5.2 RUN-WIDE PEAK MEMORY

• Peak GPU memory falls by 83.2% on average across all 21 matched comparisons under partitioned execution.Run-wide peak CPU memory is lower in 15 comparisons by 14.8% on average and higher in six by 6.8%, so it does not decrease uniformly.Source: Table 8 · p. 26, §B.5.2 · p. 26
• On Reddit, mean peak GPU memory ranges from 0.04 GB for UniGNN to 7.45 GB for SCCNN; on OGBN Products, TopoTune uses 1.46 GB and SCN 5.42 GB.These absolute Cluster-TNN measurements show that requirements depend on sampled graph context and model configuration, not input graph size alone.Source: Table 9 · p. 27, §B.5.2 · p. 27

B.6 STRUCTURAL EXPOSURE AND STATISTICAL EFFICIENCY

• Partitioned training reaches its selected checkpoint in fewer epochs in 19 of 21 comparisons but uses more optimizer updates in 20.Because each partitioned epoch contains multiple updates, epoch counts alone do not establish greater optimization efficiency; interpret them alongside updates, elapsed time, and predictive performance.Source: §B.6 · p. 28, Table 10 · p. 28

B.7 SOLUTION PATHS AND PRACTICAL GUIDANCE

• Computational costs change by workflow stage, so remedies should target the binding bottleneck rather than assume a uniform savings.Batch-before-lift applies when graph-to-domain lifting works on induced subgraphs, but repeated local lifting and altered structural exposure are trade-offs; larger cluster batches can expose more cross-cluster structure while using more memory. Report structural growth, phase-wise time and peak memory, end-to-end runtime, and predictive results.Source: §B.7 · pp. 29–30

C STRUCTURAL RECONSTRUCTION

• The recovery analysis fixes a graph partition and assumes independent reshuffling, all K clusters active, and q | K; its compatibility condition can fail for cycle-basis lifting.It holds for identity, evaluated k-hop neighborhood, and clique-complex liftings; restricting the graph can change the selected cycle basis.Source: C STRUCTURAL RECONSTRUCTION · p. 30, §C.1 · p. 30

C.2 PER-EPOCH RECOVERY PROBABILITY

• A global structure is recovered in an epoch only if its supporting clusters share a mini-batch; structures spanning more than q clusters are unobservable.Structures contained within one cluster are recovered every epoch, while recovery probability decreases with cluster span when q < K.Source: §C.2 · pp. 31–32

C.3 REPEATED STRUCTURAL EXPOSURE

• Under compatible lifting and independent reshuffling, a structure’s exposure count is binomial across epochs, and cumulative coverage converges to the set with cluster span at most q.Structures within one cluster appear every epoch, those spanning more than q clusters never appear, and intermediate spans are exposed less often; structures’ exposures within an epoch can be dependent.Source: §C.3 · p. 32, §C.4 · p. 33, §C.5 · pp. 33–34

C.6 REPEATED-EXPOSURE STRUCTURAL COVERAGE

• For any fixed exposure threshold m, expected repeated-exposure coverage converges, although at finite T expected exposure counts differ by cluster span.Source: §C.6 · pp. 34–35

C.7 MEAN STRUCTURAL EXPOSURE RATE

• Φq retains span-dependent exposure imbalance as thresholded coverage approaches its limit; the corresponding average frequency among training steps follows from M optimization steps per epoch.Source: §C.7 · p. 35

C.8 EXPOSURE EPOCH DISTRIBUTION

• For 0 < ps < 1, the epoch of a structure’s m-th exposure follows a negative-binomial distribution; larger observable spans reach the threshold later on average, while structures with cs > q never do.A one-cluster structure reaches its m-th exposure at epoch m.Source: §C.8 · pp. 35, 37

C.9 DIRECT-INCIDENCE MESSAGE AVAILABILITY

• For direct incidences, either message direction is available exactly when the higher structure is recovered, but conditional availability differs by direction and higher structures with span above q are unavailable.Every lower-to-higher message is present when its higher target is updated; a higher-to-lower message requires the additional clusters supporting the higher structure.Source: §C.9 · pp. 37–38

C.10 REPEATED DIRECT-INCIDENCE MESSAGE EXPOSURE

• Under independent epoch reshuffling and compatible message rules, conditional message exposure has mean and variance nαµ|b given n target updates; αµ|b is one for lower-to-higher messages and may remain small for higher-to-lower messages.For higher-to-lower messages, the conditional rate depends on the additional cluster span of the higher structure.Source: §C.10 · pp. 38–39

C.11 LAYERWISE INCIDENCE AGGREGATION

• For fixed inputs and parameters, conditional mean aggregation reflects message availability, but this layerwise result does not establish unbiasedness or convergence of the complete nonlinear network.Lower-to-higher aggregation has unit availability when the higher target is updated; higher-to-lower contributions are scaled in conditional expectation by the extra clusters required by each coface.Source: §C.11 · pp. 39–40

C.12 CUMULATIVE RECOVERY ENTROPY

• As T increases, aggregate marginal cumulative recovery entropy Hq,T tends to zero, and joint recovery of the observable structures occurs with probability tending to one.Source: §C.12 · pp. 40–41

C.13.1 STRUCTURAL RECOVERY ACROSS DATASETS

• Recovery is measured over 200 reshuffled epochs: q = 1 fixes recovery to structures within clusters, whereas q = K puts all nodes in one batch and guarantees complete support recovery.The diagnostic uses fixed METIS partitions and reference hyperedges, cycle-basis cells, and triangle cells; cellular recovery tracks supports, not identical local-basis selection.Source: §C.13.1 · p. 41, Figure 6 · p. 42
• At q = 1, cellular/simplicial recovery was 83.3%/86.4% on Cora Full and 97.7%/99.0% on Amazon Ratings.Neighbourhood hyperedge recovery at q = 1 was 59.7% on Cora Full and 90.0% on Amazon Ratings.Source: §C.13.1 · p. 41, Figure 6 · p. 42
• At q = 4, both 2-cell families exceeded 99% recovery on Cora Full, while all three families exceeded 99% on Amazon Ratings; Cora Full neighbourhood hyperedges reached 91.1%.Source: §C.13.1 · p. 41, Figure 6 · p. 42
• At q = 20, Questions recovery was 22.0% for cellular and 31.9% for simplicial supports despite observability; at q = 100, simplicial recovery exceeded 99.9%, while cellular recovery was 64.0%.At q = 100 on Coauthor Physics, cellular and simplicial recovery was 59.3% and 58.9%; at q = 500, it was 97.0% and above 99.9%.Source: §C.13.1 · p. 41, Figure 6 · p. 42
• At q = 1, 53.8% of Questions neighbourhood hyperedges were within single clusters, versus 3.0% of cellular and 0.9% of simplicial 2-cells; support size and partition distribution both matter.On Coauthor Physics at q = 1,000, both 2-cell families exceeded 99.8% recovery, while neighbourhood hyperedges reached 62.8%.Source: §C.13.1 · p. 42, Figure 6 · p. 42
• Ensemble inference had a higher mean test metric than single-pass inference in all 21 model-dataset comparisons, but this does not show that full-graph structural recovery explains the improvement.Source: §C.13.1 · pp. 42–43

C.13.2 RECOVERY ACROSS EPOCHS ON CORA FULL

• Empirical recovery closely matched Eq. (43) for 1-hop hyperedges, triangles, and expanded cycles, while cycle-basis recovery fell below support-based predictions because induced subgraphs can change basis selection.The comparison is shown for Cora Full across q ∈ {2, 4, 8, 16}.Source: §C.13.2 · pp. 43–44, Figure 7 · p. 43
• Entropy curves measure uncertainty about cumulative support availability, not training convergence or whether a recomputed local cycle basis selects the same reference cell.The decay epoch is an analytical threshold after peak entropy, not a convergence measure.Source: §C.13.2 · pp. 43–44, Figure 8 · p. 44

• The comparison varies K and q across three datasets at fixed model hyperparameters, reporting peak CUDA-allocated memory across training and validation.Figure 9 uses separate color scales for each dataset and places the full-graph baseline in the rightmost column.Source: D ADDITIONAL RESULTS · p. 44, Figure 9 · p. 45
• At q/K = 0.125 on Cora Full, Cluster-TNN reduced peak memory by 85.4–88.7% versus the corresponding full-graph baseline; at q/K = 0.75, reductions were 24.7–29.7%.Across all seven models, peak memory increased with sampled cluster fraction; Amazon Ratings and Questions showed the same qualitative trend.Source: D ADDITIONAL RESULTS · p. 45, Figure 9 · p. 45
• For SCCNN with K = 32, raising q from 4 to 24 increased peak memory from 4.08 to 22.05 GB, compared with 30.58 GB for full-graph execution.Source: D ADDITIONAL RESULTS · p. 45, Figure 9 · p. 45

• Experiments use ten generated random splits per dataset, with five split seeds for final evaluation; splits are shared across models and between execution protocols where both are tested.Train/validation/test proportions are 70%/15%/15% for Cora Full, Coauthor Physics, and OGBN Products, and 50%/25%/25% for Amazon Ratings, Questions, and Reddit.Source: E EXPERIMENTAL PROTOCOL AND… · p. 45
• Each model-dataset protocol is tuned separately with Optuna TPESampler to maximize the best validation metric, without evaluating test data during optimization.Studies use 100 trials for Questions, Amazon Ratings, Cora Full, and Coauthor Physics, 50 for Reddit, and 30 for OGBN Products; trials run up to 300 epochs with early stopping.Source: E EXPERIMENTAL PROTOCOL AND… · pp. 45–46
• The shared model search varies learning rate, weight decay, encoder width, and dropout, with additional backbone-dropout choices for GCN, EDHNN, and TopoTune.The search includes learning rates from 10^-5 to 10^-2, weight decay from 0 to 10^-3, and encoder widths of 32, 64, or 128 channels.Source: E EXPERIMENTAL PROTOCOL AND… · p. 46
• Partitioned training jointly searches K and q, which control partition granularity and sampled graph context; dataset-specific candidate ranges were chosen through preliminary memory assessments.Larger q can expose structures spanning more clusters while increasing lifting and execution costs; node counts and connectivity also affect mini-batch size and cost.Source: E EXPERIMENTAL PROTOCOL AND… · p. 46, Table 13 · p. 46
• Final evaluations use five seeds, and OGBN Products training runs for at most 130 epochs with early stopping; test predictions use ten independent cluster recombinations.Validation and test mini-batches use the selected training q.Source: E EXPERIMENTAL PROTOCOL AND… · p. 47, Table 14 · p. 47