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Scalable Hierarchical Graph Generation via Soft Community Structure

Ahmet Tüzen, Helge Langseth, Kjetil Nørvåg

arXiv:2610.12163v1cs.LGcs.AI

TL;DR

Generating large attributed graphs from a single reference requires fidelity and generalization without independent graph samples. SCHEMA decomposes the graph into soft communities and independently generates features and edges in three stages; across the reported evaluation it combines competitive structural fidelity with low memorization and retained downstream accuracy, while scaling to large graphs.

  • Problem

    Single-reference graph generation must reproduce structural and attribute characteristics while generalizing beyond observed edges, and existing evaluation does not fully assess memorization or downstream utility.

    Source: §1 · pp. 1–2

  • Method

    SCHEMA recursively decomposes a reference graph into soft communities and independently generates features, intra-community edges, and bridge-mediated inter-community edges.

    Source: §1 · p. 2

  • Results

    Across four datasets, SCHEMA is closest among attribute-generating models to the reference Inter/Intra ratio, reproduces only a small fraction of reference edges, and retains most reference downstream accuracy.

    Source: §4.2 · p. 8, §4.3 · p. 9, §5 · p. 11

  • Takeaways & Limitations

    SCHEMA's evaluation spans structural fidelity, memorization, downstream utility, and scalability, and the method completes on all four evaluated datasets.

    Source: §1 · p. 2

    Stage 2's all-pairs scoring within a leaf is quadratic in leaf size, which the modularity objective does not bound.

    Source: §3.7 · p. 7

Abstract

from arXiv · show

Generating large attributed graphs requires reproducing the topology, generating attributes jointly with the structure, and remaining scalable. Many real-world graphs exist as a single large graph, so a generative model has to generalize from the one graph it is fit on, without independent samples. We present Schema, which recursively decomposes a reference graph into a hierarchy of soft communities, assigning each node a membership distribution. Generation is then split into three stages, each trained independently: (1) synthesizing node attributes conditioned on soft memberships, (2) generating intra-community edges from local structural context, and (3) modeling inter-community connections over bridge nodes whose membership mass is distributed across several communities. No stage forms the full adjacency matrix, and each stage operates on a subgraph bounded by the community size. We also introduce an evaluation protocol that covers structural fidelity, memorization, downstream utility, and scalability. On four real-world attributed graphs, Schema recovers the balance between local and long-range structure more closely than any other model that generates attributes, while reproducing only a small fraction of the reference edges. It retains the downstream accuracy of the reference graph without raising it artificially above that level. Baselines that match its structural fidelity memorize the reference, while those with higher downstream accuracy either exceed the reference accuracy or fail to complete on the larger graphs. We measure scalability on six additional graphs with up to 10 million nodes.

1 INTRODUCTION

Generating a single large attributed graph requires learning from one reference while preserving its structure and features without copying its edges. SCHEMA addresses this with soft communities and a staged generator, alongside an evaluation protocol for fidelity, memorization, utility, and scalability.

1 INTRODUCTION

  • Single-graph generation must generalize from one reference while avoiding both edge memorization and loss of large-scale structure.Existing single-graph methods also face scalability, multi-scale fidelity, and node-feature limitations.Source: §1 · pp. 1–2
  • Evaluation for a single reference must assess fidelity, memorization, and downstream utility, since distributional metrics such as MMD require reference samples.The paper also identifies scalability as a limitation of existing approaches.Source: §1 · p. 2
  • SCHEMA recursively decomposes the reference into soft communities and generates features, intra-community edges, and bridge-mediated inter-community edges conditioned on shared memberships.Its stages specialize in dense within-community and sparse between-community structure.Source: §1 · p. 2
  • SCHEMA's evaluation protocol covers structural fidelity, memorization and generalization, downstream utility, and scalability; its empirical study compares nine baselines across attributed graphs up to 10 million nodes.Among attribute-generating models, SCHEMA is reported as competitive on structure, low in memorization, and able to complete on every graph.Source: §1 · p. 2

2 RELATED WORK

Prior graph-generation work is divided between generating samples from graph collections and modeling a single observed graph. SCHEMA's related-work positioning emphasizes recursive soft communities, dedicated inter-community generation, and evaluation beyond structural fidelity.

2 RELATED WORK

  • The related-work review organizes prior research around graph collections, single-graph generation, hierarchical methods, and evaluation practice.The overview distinguishes methods by their data setting and generation objective.Source: §2 · p. 2

2.1 GENERATION OVER GRAPH COLLECTIONS

  • Methods for graph collections learn distributions from independent training graphs, whereas single-graph generation aims to reproduce reference properties while generalizing beyond observed edges and identities.Collection-based methods assume many graph instances and are often demonstrated on graphs with only a few thousand nodes.Source: §2.1 · p. 3

2.2 GENERATION FROM A SINGLE GRAPH

  • Single-graph methods include classical models, embedding-based baselines, and attribute-generating approaches such as GraphMaker, GenCAT, and SynGen.The cited single-graph methods vary in whether they generate features and how they model or align features with structure.Source: §2.2 · p. 3

2.3 HIERARCHICAL AND SUBGRAPH-BASED GENERATION

  • SCHEMA differs from hierarchical and subgraph methods by using recursive decomposition, soft rather than single-label memberships, and a dedicated stage for inter-community edges.Other methods described here partition graphs or link independently generated partitions.Source: §2.3 · p. 4, Figure 1 · p. 4

2.4 EVALUATION OF GRAPH GENERATION

  • Graph-generation evaluation uses different practices across settings, including MMD for graph collections and direct statistics, homophily, transfer utility, and memorization measures for single graphs.Related work also studies trade-offs between triangle-density reproduction and overlap with the reference.Source: §2.4 · p. 4

3 METHODOLOGY

SCHEMA builds a hierarchical decomposition with soft memberships, then independently generates node features, intra-community edges, and inter-community edges. Its design bounds work by community size, though leaf size leaves a quadratic generation cost for intra-community edge scoring.

3.1 OVERVIEW

  • The decomposition returns a cluster tree with soft assignments and pooled connectivity and features, which the generator uses to produce features and both kinds of edges in one traversal.The objective is a generated attributed graph on the reference node set that reflects the reference's characteristics.Source: §3.1 · p. 4, §3.2 · p. 4

3.2 PRELIMINARIES AND PROBLEM STATEMENT

  • The formulation requires structural fidelity, joint conditioning of attributes and edges, and generalization; it preserves reference node identities and generates inter-community edges only between bridge nodes.Edge pairs are scored independently of previously sampled edges, and inherited identities mean node-label distributions are not modeled.Source: §3.2 · p. 5, §3.2.1 · p. 5

3.3.1 LEAF PARTITION AND CLUSTER TREE

  • Leiden forms leaf communities, which are contracted into supernodes and recursively partitioned into a cluster tree, with size-balanced splits used in specified fallback cases.The illustrated branching cap is b = 3, and the resulting tree may be unbalanced.Source: §3.3.1 · p. 5, Figure 2 · p. 5

3.3.2 SOFT ASSIGNMENTS VIA PROPAGATION

  • Soft assignments represent each node as a distribution over child clusters, with border nodes carrying membership across clusters; propagation updates these assignments with row normalization.The propagation is parameter-free, reuses the decomposition across training runs, and computes products over the edge list rather than forming a dense matrix.Source: §3.3.2 · p. 6

3.3.3 BRIDGE NODES AND POOLED REPRESENTATIONS

  • Bridge nodes exceed a membership threshold in another child cluster, while pooled representations summarize child features and connectivity; intra- and inter-community edges are assigned to complementary stages.Inter-community edges belong to the lowest common ancestor of their leaf communities, so no edge is modeled twice.Source: §3.3.3 · p. 6, §3.3.4 · p. 6

3.4 STAGE 1: NODE FEATURE GENERATION

  • Stage 1 generates each leaf's node features conditioned on soft assignments and pooled features, using a Transformer that jointly generates member features with masked reconstruction training.The reconstruction loss is selected by feature type: Huber for standardized continuous features and binary cross-entropy for binary features.Source: §3.4 · p. 6

3.5 STAGE 2: INTRA-COMMUNITY EDGE GENERATION

  • Stage 2 predicts candidate edges within each leaf using a variational graph autoencoder with degree-aware embeddings and a graph-convolutional encoder.Training samples edges and non-edges, while generation draws latent embeddings from the prior and selects edges under per-node degree budgets.Source: §3.5 · pp. 6–7

3.6 STAGE 3: INTER-COMMUNITY EDGE GENERATION

  • Stage 3 restricts inter-community edge prediction to a fixed-size candidate pool and adds a hierarchy-derived cluster-affinity term to bilinear pair scores.The affinity term uses the soft assignments and pooled connectivity and has no learned parameters.Source: §3.6 · p. 7

4 EXPERIMENTS AND RESULTS

Across four attributed graphs, SCHEMA is closest among attribute-generating models to the reference inter/intra-community edge ratio, keeps memorization low, and retains most reference accuracy without simplifying the task. The evaluation also examines structural fidelity, memorization, downstream utility, and scalability.

4 EXPERIMENTS AND RESULTS

  • The evaluation covers structural fidelity, memorization, downstream utility, and scalability, with structural metrics spanning local, higher-order, and mesoscopic scales.The structural metrics are degree distance, assortativity difference, triangle count ratio, and inter-community to intra-community edge ratio.Source: §4 · p. 7, §4.2 · p. 8

4.1 DATASETS, BASELINES, IMPLEMENTATION, AND REPORTING

  • The full evaluation uses four citation and co-purchase graphs, while six larger graphs are used for scalability; baselines include classical, topology-only, attributed, and subgraph-based methods.The experiments use one machine, a 12-hour limit per experiment, and report means and standard deviations over 10 generated graphs unless otherwise stated.Source: §4.1.1 · p. 8, §4.1.2 · p. 8, §4.1.3 · p. 8

4.2 STRUCTURAL FIDELITY

  • Among attribute-generating models, SCHEMA is closest to the reference Inter/Intra ratio across all four datasets, but has a larger degree distance.Its structural gain is at the community scale, not the level of individual nodes.Source: §4.2 · p. 8, Table 1 · p. 9
  • High triangle ratios coincide with the highest overlap, and pairwise edge scoring cannot reach real-graph triangle density at low overlap without copying many reference edges.The authors advise reading triangle ratio together with memorization results.Source: §4.2 · p. 8, §4.3 · p. 9

4.3 MEMORIZATION

  • The memorization evaluation measures reference-edge overlap, similarity between generated samples, and per-node overlap of generated and reference neighborhoods.These measures require node correspondence; models without correspondence are excluded.Source: §4.3 · p. 9
  • SCHEMA reproduces a small fraction of reference edges, with reference overlap at or below its overlap between generated samples on every dataset.SaGess nearly memorizes the reference, while CELL and NetGAN show partial memorization tied to their preset-overlap stopping criterion.Source: §4.3 · p. 9

4.4 DOWNSTREAM UTILITY

  • SCHEMA retains most reference accuracy on every dataset, and its TSTS-TRTR difference is negative throughout, indicating no task simplification.Its Pearson correlation is high across datasets, while architecture rankings are only partly preserved by moderate Spearman correlation.Source: §4.4 · p. 10

4.5 SCALABILITY

  • Stage 1 dominates training time on larger graphs, Stage 3 on graphs with many inter-community edges, and generation is inexpensive relative to training.GPU memory depends on the largest leaf; default generation fails for IGB Medium because its largest community exceeds one million nodes and does not fit, while capping leaves at 100,000 changes the partition.Source: §4.5 · pp. 10–11, Figure 3 · p. 11

5 CONCLUSION

SCHEMA generates a large attributed graph from one reference using recursive soft-community decomposition and three independently trained stages. It is competitive on structural fidelity while keeping memorization low and retaining downstream accuracy, with scalability to 10 million nodes requiring a leaf-size cap.

5 CONCLUSION

  • SCHEMA recursively decomposes a single reference graph into soft communities and independently generates attributes, intra-community edges, and inter-community edges.Among attribute-generating models, it is competitive on structural fidelity, keeps memorization low, and retains most reference downstream accuracy without making the task easier.Source: §5 · p. 11
  • Generation reaches 10 million nodes on a single GPU, but that graph requires capping the leaf size.Source: §5 · p. 11
  • At low overlap, pairwise edge scoring produces fewer triangles than the reference; proposed directions include triangle-count conditioning and extensions to heterogeneous or dynamic graphs.Source: §5 · p. 11

A.1 DATASETS

The appendix describes four binary-feature evaluation graphs and six larger scalability graphs with continuous features. Their domains, labels, connectivity, and component structure vary across datasets.

A.1 DATASETS

  • The four evaluation graphs are undirected, have binary bag-of-words features, and assign one class label per node.Citeseer and Cora ML represent papers and research areas; Amazon Photo and Amazon Computers represent co-purchased products and product categories.Source: §A.1 · p. 16
  • The six scalability graphs have continuous features, dense except on Flickr, and include image, social, co-purchase, financial, and citation networks.Yelp is the only multi-label graph; IGB Medium uses language-model embeddings of paper abstracts.Source: §A.1 · p. 16
  • Many connected components make the hierarchy harder because modularity gains nothing by merging nodes across components.Only 63.7% of Citeseer nodes are in its largest component, while 16.4% lie in components of at most two nodes.Source: §A.1 · p. 16

A.2.1 CLASSICAL MODELS

The classical baselines generate edge sets without node attributes and vary in how much reference information they use. The appendix describes their fitting, edge-count behavior, and an exact linear-time DC-SBM sampler.

A.2.1 CLASSICAL MODELS

  • The classical random graph models and adapted graph autoencoders generate edges only, so they serve as structural and memorization baselines rather than attribute-generation comparisons.Source: §A.2.1 · p. 16
  • Baseline information ranges from node count and density for ER G(n, p) to the full degree sequence for CL and Config.WS and BA use rounded mean-degree or attachment parameters; WS fixes its rewiring probability at 0.1.Source: §A.2.1 · p. 17
  • The DC-SBM assigns Poisson edge means using node degree parameters and block-pair edge budgets; its fitted block budgets equal observed between-block counts, or twice the internal count.Source: §A.2.1 · p. 17
  • Sampling one Poisson count per occupied block pair and then drawing endpoints costs O(M), rather than O(N^2), and is exact.The resulting multigraph is projected to a simple undirected graph, as for Config.Source: §A.2.1 · p. 17
  • The DC-SBM uses a Leiden partition at resolution 1.0, not node labels, and shares SCHEMA’s communities while replacing its learned edge model and hierarchy with a Poisson block model.Source: §A.2.1 · p. 17
  • Several classical models miss the reference edge count: Config loses edges when loops and duplicates are removed, while DC-SBM loses edges when its multigraph is projected to a simple graph.The reported degree-ratio column shows the resulting deviations; these edge counts are not corrected.Source: §A.2.1 · p. 17

A.2.2 GRAPH AUTOENCODERS

The GAE and VGAE baselines use two-layer graph convolutional encoders with inner-product decoders, with adjustments to the KL objective and edge selection for generation.

A.2.2 GRAPH AUTOENCODERS

  • Both GAE and VGAE use two-layer graph convolutional encoders and inner-product decoders, trained on positive edges and an equal number of negative edges.Both read node features and decode only edges.Source: §A.2.2 · p. 17
  • Dividing the KL term by the number of nodes prevents it from dominating the objective and collapsing the posterior to the prior.Without normalization, the latents carry almost no information about the graph.Source: §A.2.2 · p. 17
  • The decoder selects the M highest-scoring node pairs after a Gumbel perturbation at temperature 0.2, fixing the generated edge count.It receives the reference node count N and edge count M, as in ER G(n, m).Source: §A.2.2 · p. 18
  • GAE can only reconstruct the encoded reference, while VGAE supports reconstruction and prior sampling; only prior sampling is reported for VGAE.GAE defines no prior.Source: §A.2.2 · p. 18

A.2.3 PUBLISHED IMPLEMENTATIONS

The baseline implementations generally use released code and published settings, with several consequential choices or exclusions shaping how their results should be read.

A.2.3 PUBLISHED IMPLEMENTATIONS

  • The remaining baselines use released code and published hyperparameters, with reported deviations or implementation-forced choices; SaGess has no deviations.SynGen rounds node count to a power of two and generates structure and features independently; LGSG’s node assembler targets the reference node count but may miss it, and does not control edge count.Source: §A.2.3 · p. 18
  • CELL and NetGAN stop at preset reference-edge overlap, so their memorization results reflect the stopping rule rather than generalization.The published stopping criteria are used on every dataset.Source: §A.2.3 · p. 18
  • NetGAN generates only the largest connected component, so its size-dependent metrics use a smaller graph than other methods’ results.Its results are computed against the reference graph’s largest connected component.Source: §A.2.3 · p. 18
  • GenCAT receives the reference class-mixing matrix, class-wise attribute statistics, and degree sequence, while its continuous, nearly dense features make binary-reference marginal comparisons uninformative.Its low degree distance partly reflects the supplied degree sequence.Source: §A.2.3 · p. 18
  • GraphMaker is label-conditional, and its released implementation stores a dense N × N adjacency matrix that limits graph size.The main text reports its asynchronous variant; the appendix reports both variants.Source: §A.2.3 · p. 18
  • CGT is excluded because it generates a separate computation graph per node rather than one graph, leaving the structural and attribute metrics undefined.SANGEA is discussed but not experimentally included because no implementation is publicly available.Source: §A.2.3 · p. 18

A.3 FULL STRUCTURAL FIDELITY

Structural fidelity is assessed across local, higher-order, mesoscopic, and global scales, and the reported results show that close metric matches can carry important trade-offs.

A.3 FULL STRUCTURAL FIDELITY

  • Classical ER, CL, and Config models have Inter/Intra ratios from 2.4 to 20.5, while CL and Config match degree distributions but lose community structure.DC-SBM is the only classical model given a community partition, so degree-based modeling alone does not recover the mesoscopic scale.Source: §A.3 · p. 20
  • DC-SBM is the strongest classical baseline, with Inter/Intra ratios from 0.80 to 1.13 and the lowest modularity difference on three datasets, but it receives the reference Leiden partition.SCHEMA starts from the same partition and reaches a comparable ratio on four evaluation graphs with a learned edge model while also generating attributes.Source: §A.3 · p. 20
  • GenCAT has the lowest degree distance across datasets and completes on all five, but its Inter/Intra ratio is 1.7–3.0 on four, misplacing edges between communities.On Flickr, degree distance is 0.020 and assortativity difference is 0.002; it receives class-mixing, attribute, and degree information from the reference.Source: §A.3 · p. 20
  • GAE’s degree ratio is exactly one by construction, but its triangle ratios reach 579 on Flickr; VGAE has Inter/Intra ratios of 9.9 on Citeseer and 13.0 on Amazon Photo.GAE produces triangle ratios of 11.6, 6.1, and 9.0 on three other datasets, placing its matched edge count in dense clusters.Source: §A.3 · p. 20
  • Completion falls with scale: all models run on the citation graphs, while one fails on Amazon Photo, five on Amazon Computers, and six on Flickr.Failures are attributed either to memory exhaustion or time limits; the failing models produce a full score matrix or use diffusion.Source: §A.3 · p. 20
  • SaGess is close to the reference on most reported columns but reproduces most reference edges, while LGSG’s merging discards edges on Amazon Photo and Cora ML and overshoots on Citeseer.CELL matches edge count exactly and NetGAN is at 0.85 of it where it runs; SynGen matches edge count but rounds node count up to a power of two, producing degree ratios from 0.68 to 0.93.Source: §A.3 · pp. 20–21
  • SCHEMA stays near the reference degree and Inter/Intra ratios on two citation graphs and Amazon Photo, with characteristic path length ratios of 0.83–0.98 across four datasets.Its triangle ratios across those four datasets range from 0.28 to 0.55, consistent with independently scoring node pairs.Source: Table 7 · p. 21, §A.3 · p. 21
  • On Flickr, SCHEMA’s 500-node candidate pool yields degree distance 2.287 but Inter/Intra ratio 0.097; enlarging it to 4000 raises that ratio to 0.414 while degree distance rises to 10.772.A 2000-node pool raises the Flickr triangle ratio to 122.9; the reference has 63,941 triangles over 89,250 nodes, making the ratio sensitive to small absolute excesses.Source: §A.3 · p. 21

A.4 MEMORIZATION EVALUATION

With a single reference graph, edge overlap is expected, so memorization requires interpretation beyond overlap alone.

A.4 MEMORIZATION EVALUATION

  • In the single-reference setting, some edge overlap is expected because the model is trained to produce a graph resembling that reference.The multi-graph definition of memorization as isomorphism or near-isomorphism to a training graph does not apply here.Source: §A.4 · p. 21
  • Non-zero edge overlap alone does not establish memorization, since independently generated edges can still share edges when degree sequence and community structure are reproduced.Source: §A.4 · p. 21
  • Self-overlap measures mean overlap between samples from the same model; high self-overlap with low reference overlap signals nearly identical outputs across samples.Source: §A.4 · p. 22

A.4.1 ALIGNMENT

The section evaluates alignment-aware memorization, downstream classification, scalability, and generation at variable graph sizes, alongside graph-size scaling and ablations of Schema’s stages and components. It reports structural overlap and fidelity results, including how noise, partitioning, and cluster affinity affect them, and notes limits on interpreting some memorization and utility measures.

A.4.1 ALIGNMENT

  • Memorization comparisons require node correspondence: arbitrary-identity models are excluded, degree-aligned models are matched by degree, and node-aligned models retain reference identities.The attribute metrics use nearest- and second-nearest-reference distances for NNDR and a threshold-based leak fraction, supplemented by attribute density and frequency correlation; structural measures include edge, self-, and neighborhood overlap.Source: §A.4.1 · p. 22, §A.4.2 · p. 23

A.4.3 RESULTS

  • GAE’s edge overlap is 0.008–0.227 but self-overlap is 0.743–0.951; SaGess reaches 0.910 and 0.815 edge overlap on Citeseer and Cora ML, while CELL’s citation-graph neighborhood Jaccard is about 0.5.Degree-aligned models score below 0.03 on every measure and dataset, while none has a memorized node; low overlap alone therefore does not establish good generation.Source: §A.4.3 · pp. 23–24, Table 10 · p. 24
  • SCHEMA’s edge overlap is 0.003–0.102, its self-overlap exceeds edge overlap on all five datasets, and its memorized-node fraction is at most 0.034 on four datasets, versus 0.183 on Citeseer.On Citeseer, 82.9% of SCHEMA’s memorized neighborhoods come from leaf communities of at most two nodes; degree-preserving rewired graphs reach a memorized-node fraction of 0.286 ± 0.003.Source: Table 10 · p. 24, §A.4.3 · pp. 24, 26
  • SCHEMA’s attribute-frequency correlation is 0.858–0.981, and its leak is at most 0.002 except on Flickr; there, its leak is uninterpretable because its attributes have smaller norms than the reference.NNDR ranges from 0.969 to 1.000 across all models and datasets, so it does not discriminate in this sparse, high-dimensional binary setting. Leak is interpretable only when generated attribute marginals match the reference.Source: §A.4.3 · pp. 24–25, Table 11 · p. 25

A.5.1 ABSOLUTE ACCURACIES

  • On Flickr, the reference accuracy is 0.50 against a majority rate of 0.42, while every generated graph returns exactly the majority rate under TSTR and TSTS; SynGen falls below the majority rate on all four evaluation datasets.Flickr does not separate models, so no further downstream results are reported for it. Utility is defined as TSTR divided by TRTR, averaged over four two-layer architectures and five seeds.Source: §A.5.1 · p. 26, §A.5 · p. 26, Table 12 · p. 27

A.5.2 HOMOPHILY AND THE TSTS GAP

  • SCHEMA’s generated homophily is 0.16–0.25 below the reference and feature-only accuracy is 0.16–0.32 below; its TSTS is 0.07–0.13 below TRTR and comes within 0.05 when rewired to reference homophily.For generated graphs that are easier to classify, homophily is also higher than the reference; matching it by rewiring leaves a difference of −0.033 to +0.005. SCHEMA’s unexplained remainder ranges from −0.001 on Citeseer to −0.084 on Amazon Photo.Source: §A.5.2 · pp. 27–28, Table 14 · p. 28

A.6 SCALABILITY

  • SCHEMA’s clustering and training take up to 8.2 hours on OGBN Products, while generating one graph takes one to seven minutes on Yelp, OGBN Products, and IGB Medium and under 30 seconds on the other graphs.One graph costs 0.5%–2% of clustering and training time on every graph; GenCAT exhausts host memory on OGBN Products and IGB Medium.Source: Table 15 · p. 29, §A.6 · p. 29

A.7 GENERATING GRAPHS OF ANY SIZE

  • SCHEMA changes graph size by replicating or subsampling communities, with target node count sN; independent noise is used in Stages 1 and 2, and Stage 3 reassigns edges across copies while preserving node degrees.Duplicate reassigned edges are dropped, reducing edge count; the loss is greatest at scale factors 2 and 2.5.Source: §A.7 · pp. 29–30
  • At scale 100, generation completes with 8.9M nodes and 37.3M edges, while the edge ratio against the reference stays near the scale-one value of 0.83.For scale factors above one, the generated-to-scale-one edge ratio stays within 1.5% of one and approaches one as scale grows; at scale 0.5 it is 1.040 because the selected subset exceeds the target node count.Source: §A.7 · p. 30

A.8.8 RESULTS

  • With deterministic Stage 2, switching node noise on changes degree distance from 1.502 to 0.590 and triangle ratio from 1.171 to 0.417; when Stage 2 generates, both changes are under 0.01.Node noise affects structure through the encoder: Stage 2’s prior does not read attributes, so Stage 1 has little structural effect once that prior is used.Source: §A.8.8 · p. 32

A.8.2 PARTITIONING

  • Random communities of Leiden’s sizes raise degree distance from 1.075 to 4.026, while Louvain lowers it to 0.933 but raises Inter/Intra ratio to 1.263.Random partitioning retains the same sizes but assigns nodes without structural meaning; its triangle ratio remains 3.515, as does Leiden’s. The ablation replaces Leiden in both the leaf partition and recursive tree construction.Source: §A.8.2 · pp. 30–31, §A.8.8 · p. 32

A.8.7 CLUSTER AFFINITY

  • Without the cluster-affinity term, Inter/Intra ratio rises from 1.030 to 1.247 and degree distance rises to 1.432; using it as a KL target recovers part of the change.In the reported setting, the hierarchy-derived term is added to pair scores with a learned weight; the alternative KL-target setting does not add it to generation scores.Source: §A.8.7 · p. 31, §A.8.8 · p. 32

A.8.3 HARD ASSIGNMENT

  • Hard assignments yield an Inter/Intra ratio of 0.838 versus 1.030 with soft assignments, while binary and single-level hierarchies stay within 0.09 of the reported ratio.One-hot assignments remove bridge nodes; the binary tree raises degree distance to 1.365. Replacing Stage 1’s Transformer with an MLP keeps degree distance, triangle ratio, and downstream gap close to reported values, whereas Stage 3 attention yields an Inter/Intra ratio of 0.827 and triangle ratio of 0.181.Source: §A.8.3 · p. 31, §A.8.4 · p. 31, §A.8.6 · p. 31

A.8.6 STAGE ARCHITECTURES

  • Stage 2’s inner-product decoder and MLP predictor improve triangle ratios to 1.042 and 1.196, but raise edge overlap to 0.071 and 0.079 and self-overlap to 0.127 and 0.131.The reported configuration’s triangle ratio is 0.275; the simpler decoders recover triangle density by staying closer to the reference edge set.Source: §A.8.6 · p. 31, §A.8.8 · p. 32

A.8.5 JOINT ATTRIBUTE GENERATION

  • Replacing generated attributes with reference attributes changes structural metrics by at most about two standard deviations, since neither edge stage reads generated attributes at generation.Stage 2 draws latents from the prior, and Stage 3 reads memberships and pooled connectivity; attributes and edges are linked through shared soft memberships, not realized attributes.Source: §A.8.5 · p. 31, §A.8.8 · p. 32
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