GraphUniverse: Synthetic Graph Generation for Evaluating Inductive Generalization
Authors: Louis Van Langendonck
Guillermo Bernardez
Nina Miolane
Pere Barlet-Ros
Venue: ICLR 2026 Poster (2026)
Keywords: Graph Neural Networks
Synthetic Dataset Generation
Graph Benchmarking
Primary area: datasets and benchmarks

Abstract: A fundamental challenge in graph learning is understanding how models generalize to new, unseen graphs. While synthetic benchmarks offer controlled settings for analysis, existing approaches are confined to single-graph, transductive settings where models train and test on the same graph structure. Addressing this gap, we introduce GraphUniverse, a framework for generating entire families of graphs to enable the first systematic evaluation of inductive generalization at scale. Our core innovation is the generation of graphs with persistent semantic communities, ensuring conceptual consistency while allowing fine-grained control over structural properties like homophily and degree distributions. This enables crucial but underexplored robustness tests, such as performance under controlled distribution shifts. Benchmarking a wide range of architectures—from GNNs to graph transformers and topological architectures—reveals that strong transductive performance is a poor predictor of inductive generalization. Furthermore, we find that robustness to distribution shift is highly sensitive not only to model architecture choice but also to the initial graph regime (e.g., high vs. low homophily). Beyond benchmarking, GraphUniverse’s flexibility and scalability can facilitate the development of robust and truly generalizable architectures. An interactive demo is available at https://graphuniverse.streamlit.app.

Published as a conference paper at ICLR 2026
GRAPHUNIVERSE: SYNTHETIC GRAPH GENERATION FOR EVALUATING INDUCTIVE GENERALIZATION
Louis Van Langendonek
Guillermo Bernárdez
Universitat Politècnica de Catalunya, Spain louis van langendoncc @upc. edu
University of California Santa Barbara, USA gui 1lllmo_ebbbe www.eeeeeeeeeeeeeee. edu
Nina Miolane
Pere Barlet-Ros
University of California Santa Barbara, USA ninami aaaaaneuusst edu
Universitat Politècnica de Catalumya, Spain pere bar.ettuup..ed
ABSTRACT
A fundamental challenge in graph learning understanding how models generalize to new, unseen graphs. While synthetic benchmarks offer controlled settings for analysis, existing approaches are confined to single-graph, transductive settings where models train and test on the same graph structure. Addressing this gap, we introduce Graphnniverse, a framework for generating entire families of graphs to enable the first systematic evaluation of inductive generalization at scale. Our core innovation is the generation of graphs with persistent semantic communities, ensuring conceptual consistency while allowing fine-gre ime-grained control over structural properties like homophily and degree distributions. This enables crucial but under- explored robustness tests, such performance under controlled distribution shifts. GNNs to graph transformers
Benchmarking a wide range of architectures- and topological architectures- reveals that strong transductive performance is a poor predictor of inductive generalization. Furthermore, we find that robustness distribution shift is highly sensitive not only model architecture choice but also to the initial graph regime (e.g., high vs. low homophily). Beyond bench- marking, Gripeeniverse's flexibility and scalability can facilitate the development of robust and truly generalizable architectures. The framework is open-source at btttps:/gitte wom http://hhh....ooooooooeeeaaaeeaa
INTRODUCTION
Graph learning has emerged as powerful paradigm for learning from relational data across diverse domains, from drug discovery (Wong et al.] 2024) and fraud detection (Cheng et al.) (025) to knowledge graphs (Galkin et al. (023). Graph Neural Networks (GNNs) (Scarselli al.) 2008), with
its countless variants (Gilmer et al.] [217]] Kipf & Welling 2017 Hamilton et al.) 201 2017 Xu et al.
2019: Veličković et al. 2018), have demonstrated remarkable success extending deep learning frameworks to graph-structured data, achieving competitive performance on tasks ranging from node classification graph-level prediction. However, unlike the transformative leap from task-specific models to general-pune purpose architectures observed in natural language processing and computer vision, graph learning remains largely limited to specialized, task-specific models with limited evidence of robust generalization and scaling capabilities.
Recent analyses argue that progress in graph learning is hindered by a flawed benchmarking culture. Bechler-Speicher et al. (2025), for instance, critiques the field's excessive focus incremental gains on weak benchmarks, which often fail to outperform simpler non-graph baselines. They also highlight scarcity of large-scale, diverse datasets, arguing that these limitations hinder the development of models that can generalize and scale. Complementing this, Wang et al. (2025) pinpoints critical gaps in the theoretical understanding of model behavior-parriculatt particularly concerning robustness to distribution shifts and generalization guarantees. They identify these theoretical weaknesses as key obstacles preventing graph models from advancing beyond narrow, task-specific applications.

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To remedy these issues, both works propose creating better datasets through synthetic generation and quality-centric curation, alongside developing metrics for generalization, robustness, and trustwor- thiness. However, existing synthetic generation tools like GraphWorld (Palowitch et al., 2022) fundamentally limited in this regard. They generate graphs as isolated, independent instances, which restricts evaluation transductive settings where model trains and tests the same structure. This single-graph paradigm makes it impossible to study generalization to unseen graphs and constrains to experiment at scale- scale-precisely the two capabilities identified as critical for building powerful graph foundation models (Wang et al. 2025).
We address this gap with GraphUniverse a framework for generating graph families at scale. Our contributions can be summarized as follows:
1. We develop hierarchical generative model extending Degree Corrected-Stochasie Block Models (DC-SBMs) (Karrer & Newman 2011) to an inductive setting with multiple graphs that maintain semantic consistency- node identities or community structures persist across different graph instances-while enabling controlled variation in their structural properties.
2. GraphUniverse is available with source code at https:://ittub. com/ We have integrated GraphUnivens into TopoBench (Telyatnikov et al. 2025)) to facilitate reproduction and extension of our experimental results.
PyPI package
https::/pypi..o
srrjject/graph-univess..
Loui SeauiaaVrraarggeddncc/rrree wtw.://ooccc/aaaaaainieeeeseee
We conduct systematic benchmarking comparing inductive and transductive evaluation across classical and contemporary graph architectures, revealing differrnnces in model rankings between paradigms. Additionally, we evaluate model robustness under controlled property shifts, an analysis only possible with our inductive framework, finding that robustmess strongly depends on both architecture choice and initial graph properties. These findings challenge conventional assumptions about graph model performance and demonstrate the critical importance of evalu- ation paradigm choice in assessing true model capabilities. Furthermore, we demonstrate that SaaphUniverse-generatee datasets can serve effective proxies for real-world datasets, with model rankings showing strong correlations with those obtained on real data.
4. We provide an interactive web platform (https: / ht/ggrappppt graphun wwtrhuuiiierre... verse. www.amlit..taaammm app/ for visualization, exploration, and direct download of generated datasets.
We envision Grapheniverse a versatile tool for diverse research applications, from targeted generalization benchmarks to large-scale data generation and augmentation for model pre-training. While experiments demonstrate its immediate utility, they represent only fraction of what is possible with controllable graph family generation. Thus, we release Griperniverse to the community flexible framework, inviting extensions and adaptations explore new frontiers in graph learning.
2 RELATED WORK
The evaluation of graph learning models has evolved from early, limited-scope benchmarks (Dwivedi) et al. 2023, Morris et al.) 2020) to large-scale, real-world datasets. The Open Graph Benchmark (OGB) (Hu et al.) 2020) was a significant step forward, providing standardized protocols on large graphs that revealed critical challenges in generalization. Subsequently, the GOOD benchmark (Gui et al. 2022) introduced a focus on out-of-distribution (OOD) generalization by creating splits designed test robustness to covariate and concept shifts. However, fundamental limitation of these real-world benchmarks is their static nature. The datasets are fixed, the properties of the data splits are not tunable, and as recent critiques have noted, they often lack sufficient coverage of important graph properties like heterophily, limiting their utility for systematic model analysis (Bechler-Speichee al. 2025).
Recognizing the limitations of static datasets, growing consensus advocates for high-fidelity synthetic data generation as a path toward more principled and scalable evaluation (Bechler-Speicher
al. 2025 Wang et al. 2025). The most prominent effort in this direction is Graph World (Palowitch
et al. 2022), which uses synthetic generation to study model performance across a space of graph
properties. Related efforts like CGT (Zahirnia al., 2023) and the metadata-driven approach (Li
cet al. 2023) provide valuable insights into model behavior by mapping real-worlh graphs to synthetic equivalents and analyzing performance across graph properties, respectively. While these approaches
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enable controlled analysis, they remain confined to generating independent, single graphs. This restricts evaluation to the transductive setting. where models are tested on the same graph structure seen during training, and fundamentally prevents the study of model's ability to generalize to entirely new and unseen graphs.
The importance of synthetic data extends beyond benchmarking foundation model development. GraphFM (Lachi et al. 2024) leverages Graph World-style synthetic graphs to expand its pretraining corpus, though limited transductive settings, while OpenGraph (Xia et al.) 2024) uses LLMs to augment existing datasets-mirroringg synthetic data's ' established role in computer vision (Tobin et al. 2017) and NLP (Wang al., 2023). However, existing graph generation frameworks cannot provide the inductive generalization and systematic coverage of graph modalities that robust foundation
models require Meecler-Speicchn et al. 2025 Wang 2025
capability limited multi multi-graph
approaches.
GraphUnivers directly addresses these limitations. Unlike static benchmarks, it provides a generative framework capable of producing unlimited data with fine-grained control over structural properties. Critically, unlike Graph World and other single-graph approaches, it generates entire families of graphs with shared semantic meaning, enabling systematic study of inductive generalization. Further- more, our experiments validate that Graph Univerue geneeratd data closely mirrors real-world model behavior, suggesting its potential pre-training applications. We present Graph Universe as an open-source tool that researchers can extend and adapt for their specific needs, whether for controlled benchmarking sophisticated data augmentation strategies for Graph Foundation Model developement, of which detailed discussion is provided in Appendix
complementary data source for model development, including
basis for more
BACKGROUND
Community-basee graph generation provides interpretable control over node-level properties and their relationships, naturally supports community detection tasks, and reflects the modular organization commonly observed in real- world networks (Fortunato 2010). This section revisits some previous works on this topic that Ctraphnnveene draws inspiration from.
Let G = (V. E) be an undirected graph with [V = nodes and A E {0. 10 1)"X" its adjacency ma- trix, where A:j Aja and A, = 0. Let ie E N be the number of communities and b, E (1,..., k) the community label of node i. We denote by P [0. 1]=1 the (symmetric) block/community edge probability matrix with entries Pr.. with r, (k..
Stochastic Block Model (SBM). SBMs (Holland et al. 1983) generates graph by first uniformly sampling labels by...., bn then drawing edges independently as
Ai Bernoulli(Pb.) Po.b,) (i<i<js>
SBM is used in two complementary ways; (i) the inference view: given single observed A, estimate (b, P); and (ii) the generative view (the one adopted in this work): given (n, P). sample A.
Pegree-Correeted SBM (DC-SBM). A limitation of SBM is that it enforces homogeneous ex- pected degrees within a community, since edge probabilities depend only on block membership. To address this, the original DC-SBM (Karrer & Newman 2011)) implementation introduces node- specific degree factors 0, and a nonnegative block matrix e k Moreover, they shift focus from simple graphs to multigraphs, i.e. graph with multi-edges and self-loops. its original Poisson multigraph form, edges are counts with rates:
Aig Poisson(Xc)) (i ().
Aúi 2
Poisson( 0? An, where Auj =- 0,0; Angby
Imposing a per-group sum-lo-one normalization on degree factors, i.e. Eeh, Og = -1 Vr E {1...., k), the expected number of edges between communities r and s (counting each undirected edge once) becomes:
(Ars: r r s,
E M
Aprs F = s,
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so A controls the total number of edges in block, while 8 redistributes degrees within blocks. This Poisson formulation is standard in enference-focuree works (Karrer & Newman
2011: Abbe 2018).
Generative Bernoulli Formulation. Since the original DC-SBM naturally generate Poisson multi- graphs, common approach to get simple graphs from them is to collapse multi-edges into single undirected edges and remove self-edges after generation (as done in GraphWorld (Palowitch et al. 2022)). However, this leads to a systematic but unpredictable mismatch between the imput parameters and the properties of the output graph. Therefore, we choose to work directly with a Bernoulli reformulation of the DC-SBM algorithm the basis for our generator (Rohe et al. 2018). Its justification and precise correspondence with the Poisson DC-SBM are deferred to Appendix C Moreover, discussion of the limitations of relying on DC-SBM models (e.g. lack of higher order structures) is provided in Appendix D
4 GRAPHUNIVERSE
While DC-SBMs generate individual graphs with controllable community structure, they cannot support inductive generalization studies due independent graph generation with weak community- specific signals. To address this, we introduce a hierarchical generation framework that decouples global community properties from local graph characteristics, enabling systematic investigation of model performance across semantically related graph distributions.
4.1 THREE- ARCHITECTURE
Our framework is organized hierarchically into three levels, each controlling different aspects of graph generation-See see Figure !
Universe Level (Global Community Properties). At the top level, a Graph Universe (left panel in Figure T defines master set of K persistent communities. These communities, assigned the node-level, retain stable semantic identities across all generated graphs, specified by:
Structural patterns: The universe-level edge propensity matrix E R encodes relative inter- community connection strengths. Unlike standard DC-SBMs with uniform block probabilities, we introduce heterogeneity by generating P... 1 Ers where Ers N/0 (2c)²) with variance parameter c. This perturbation is symmetrized and rescaled to preserve target homophily and degree constraints (details in App. E.. yielding fine-grate -ra-gra structural variation across community pairs.
Degree profiles:
community-speciioo
degree propensity vector ō = (51....,55 5x) E [-1,1]]
with d2 Uniform( (1 1), determines the characteristic degree propensity of each community. Unlike standard DC-SBMs where degree factors are independent of community membership, each oş anchors community ie in the degree spectrum, with 6% 1 corresponding to low-degree node tendencies and 6₁ = +1 to high-degree ones.
Feature distributions: Community centroids μ4 E Rd. for E (1..... :I.....;]. are sampled from μą ~ (N, La), with Le controlling cetween-communjiy feature separation. Node features within each community are then generated from A (JuR; within-community feature variance.
a T where Jusee controls
Family Level (Generation Constraints). A Graph Family (middle panel Figure I specifies allowed ranges for graph-level parameters while maintaining consistency through the community- behaviour defined by the universe-level identities:
Structural ranges: Target homophily h E [Amin, Amax] and average degree d E [dmins dmax).
Size constraints: Number of nodes n E [mmins [max] and number of participating communities k E [kmin, kmma] per graph, with kmax < K.
Coupling parameters: Degree separation p E [Pmins Pmaa] controls the overlap between community degree distributions (low P yields broad overlap, high P yields well-separated distributions). and degree distribution parameters such as power-law exponent a E [Ormnn Omax]
Graph Level (Instance Generation). Individual graphs are generated Graph Sample instances (right panel in Fig. [ each obtained by sampling specific values from family-level ranges and inheriting community properties from the universe. The full procedure is described in next section.
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UNIVERSE
FAMILY
A communitis
1) Sample family
(1, n,,, k,poo)
e
% 40
4
2) Select & communitie (f
edge propensity
O
homophily Ausis Proaz
3) Sample Scale P to probability mattix
Ada
drgece dasiar thece
nodie Nexin Rocas
4) Generale graph
4.3 generation
4-4 freture gemeration
degree properaaity
communitins Reuier Povse per grapa
-9
mode initinlication
dogree iepentiion
distribution clustees
10-12 12
2
0
+5
pelwer loor pxpement
4.2 & aassigmmmnt
initiaiiationn
Figure 1: Overview of GraphUniverse generation methodology.
4.2
GRAPH INSTANCE GENERATION PROCESS
Each graph instance is generated in four phases, shown in the Generate Graph section in Figure]
!
Phase Parameter Sampling. Graph-specific parameters are drawn uniformly from family ranges:
(n.k,,,d.p.,)) Uniform miii Pmmax] x ... x [[mmins Amaxx) ).
(1)
Phase 2:
Community Selection. We randomly select & communities C = (cI...., c S (l,.... K} from the universe to appear in this particular graph.
Phase 3: Probability Matrix Construction. We extract the k x k submatrix
ji = sl.,s, i E.j kk.
To obtain valid Bernoulli probabilities with the sampled graph-level properties, we rescale in two stages: (i) homophily adjusiment: apply separate scaling factors to diagonal vs. off-diagonal entries so that the within- and apply global multiplier that the mean entry matches the target edge density d/(n 1) for nodes.
ratio matches the target homophily h; (ii) density adjustment:
The resulting matrix P Pscaled preserves the heterogeneity of P sub while satisfying both constraints (see Appendix[E].
Phase 4: Graph Realization with Community Properties. The final graph is generated follows:
1. Node assignment. Nodes are distributed uniformly across the selected communities C.
2. Degree factors. For each node in community c(t), we assign a degree factor 0; by coupling degree distributions communities. We first sample power-law degree factors and sort them as (0(1), (n)) Each community's 's degree center ōe(i) maps to a preferred rank Me(()
148.1 (n 1), and we assign degree factors by sampling rank indices from (( o²)) truncated 2 to [1 nl. where 2² is determined by the degree separation parameter Full details are in Appendix
Edge generation. Each pair (i,j) with i is connected independently with probability
Ps = min (1, 0,0,P
(2)
After sampling, we verify connectivity and connect any disconnected components by adding edges that minimize deviation from the target block structure P Pscated (details in Appendix
4. Feature generation. Finally, node features are sampled from community-specife Gaussian distri- butions:
x; (N(Ho(()) o'I).
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Graph Generation Parameters
and T*" signfficncc levels shown. NS Not Sighticant
Edge Propensity Voriance c
N8
NB
0.89
Cluste Variance o*
NS
N5
NS
NS
N5
-8.61
NS
NS
-0.63 P.
N5
NS
Comalation Coofficioo
Moan Homophily
0.5
Mean Degree
N8
0.96
4..2
0.53
-0.21
N8
NS
0.54
N8
N8
N8
-0.
NS
4.61
0.24
0:45 PM
NS
Mean Communtliss Partolppting
N6
NS5
N6
0.008
-0.91
-0.77
-0.02
-0.77
0.50
Exparation
N8
NS
0:41
NS
N5
NS
NS
N6
N8
-10
Mean Power Law Exponent
NE
0.23
0..52
N8
Muan Homophly Levelo
Alean Avarage Dagine
Meen Engge Protabbllty Deviation
Mean Tall Ratio 69
Meen Graph Generation Time
Festure Signal
Degias Signal
Strustoee Signal
Festura Convitency
Degroe Commatency
Stuucure Consatercy
Validation Metrics
Figure 2: Parameter sensitivity heatmap from 100 randomized graph families with all parameters simultaneously varied across complete ranges. Pearson correlation coefficients are shown with stars indicating significance levels. NS indicates no statistically significant correlation.
4.3 VALIDATION OF GRAPHUNIVERSE
To ensure our multiple graph generation framework produces high-fidelity graphs with the intended properties and learnable signals within and across graphs, conduct comprehensive validation study into three critical aspects: graph properties, signal strength, and cross-graph consisteney.
Validation Metrics. We define three categories of validation metrics. Graph Property metrics verify that generated graphs match target structural characteristics, including homophily levels, average degree, degree distribution tails, and deviations from expected community edge probability matrices. Signal Strength metrics assess the predictability of community labels using different node-level features (node features, degree, and multi-hop neighborhood structure), ensuring graphs contain learnable signals for downstream tasks. Cross-Graph Consistency metrics evaluate whether community identities remain semantically consistent across different graph instances through feature centroid similarity, structural pattern correlation, and degree ranking preservation. Detailed definitions for all metrics are provided in Appendix
Parameter Sensitivity Analysis. We generate 100 distinct graph families-30 graphs each-with completely randomized parameter configurations sampled uniformly across broad ranges (further de- tails in Appendix H). This stress-tests the framework's 's robustness by capturing parameter interactions extreme values rather than nominal operating points. We assess parameter-metric relationships using Pearson correlations on family-level means, reporting only statistically significant correlations (p < 0.05). The correlation heatmap of Figure 2 shows the parameter (y-axis) responsiveness across all validation metrics (x-axis).
Validation Results. Graph property metrics (left panel in 2 show expected strong correlations: parameters precisely control homophily and average degree, with slight deviations under extreme settings reflecting our multiplicative edge generation process. Signal strength metrics (middle panel in in confirm theoretical expectations: cluster variance controls feature signals, homophily/degree enhance structure signals, and fewer communities simplify classification. Cross-graph consistency metries (right panel in [n provide the strongest validation of our hierarchical design, with propensity variance governing structure consistency, degree separation controlling degree consistency, and cluster variance determining feature consistency, A detailed analysis the parameter validation results, as well as individual parameter effect plots, are provided in Appendix Section
4.4 SCALABILITY
Table I: GraphUniverse generation stats.
GraphUniverse demonstrates linear scaling across graph sizes, enabling efficient large-scale evaluation. Table [ shows generation performance measured on
Avg. number
Time per graph (see)
Throughput (graphs/sec)
of nodes
10
0.002
449.7
AMD Ryzen? 7 5700U CPÚ processor (single-threaded), threaded),
100
0.023
42.8
500
0.349
2.9
averaged over 100 graphs.
1000
1.309
0.8

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4.5 INTERACTIVE EXPLORATION TOOL
To complement the open-source package, we provide an interactive Streamlit application for code- free exploration of the GraphUniverse generator, available both a hosted demo (https: / graphuni verse. ream
Seamlite lit. (ttamiittppp/) and for local deployment via the PyPI package. The app allows users define a generation universe, tune family parameters (e.g., homophily, degree distribution), and instantly visualize the resulting graphs, their properties and validation metrics. Generated graph families can be downloaded as PyTorcc Geometric InMemoryDataser objects.
BENCHMARKING
The ability generate diverse graph families with fine-grained control over their properties opens up countless avenues for systematic model evaluation. To demonstrate this potential, we present a benchmarking suite designed to probe three fundamental research questions (RQ) of inductive perfor- mance, generalization and robustness in modern GNNs. This investigation, while comprehensive, represents just one of the many possible explorations that GraphUniverse makes possible.
5.1 EXPERIMENTAL SETUP
Implementation. All experiments conducted using the TopoBench benchmarking frame- work (Telyatikkvv et al.] (2025). which we extend with a custom GraphUniverse loader to sys- tematically define and iterate evaluate models in both inductive and transductive settings on two distinct tasks: node-level com- munity detection (classification) and graph-level triangle counting (regression). Unless otherwise specified, we consider a set of fixed dataset generation parameters- succh as the number of graphs in inductive settings (1000), or the number of nodes in transductive ones (1000) to ensure a consistent baseline across experiments (defaults lists in Appendix M.T..
graph generation parameters. Across benchmarking,
Models. We evaluate diverse set of architectures representing major paradigms contemporary graph learning (a brief description of each of them can be found in Appendix [M.2]: M..2) DeepSet (Zaheer
ct al. 2017), GraphMLP (Hu et al. (2021), GCN (Kipf & Welling 2017). GraphSAGE (Hamilton
al. 2017). GIN (Xu al.] (019)) GAT v2 (Brody al.22) Topo'Tune (Papillon al. [0225).
Neural Sheaf Diffusion (Bodnar et al. 2022 and GPS (Rampášek et al. 2022) (see Appendix M.2).
Hyperparameter Optimization. For each model-dataset configuration, conduct comprehensive grid search hyperparameter optimization using architecture-specffic parameter grids detailed in the Appendix M.3 Each configuration is evaluated across different dataset instantiations generated with the same input parameters but different data random seeds, with the configuration achieving the highest mean validation performance selected for final test evaluation.
Evaluation Metries.
We report test accuracy and mean absolute error (MAE) for community detection and triangle counting tasks, respectively (averaged across 3 random data seeds with standard deviations) For both inductive and transductive experiments, we consider a 70/15/15 training/val/test split.
5.2 RQ1: DO GRAPH LEARNING MODELS PERFORM DIFFERENTLY IN THE INDUCTIVE SETTING, FUNCTION OF KEY VARIED GRAPH PROPERTIES?
Motivation. Existing synthetic graph generation models (GraphWord) World (Palowitch et al. 2022 only allow for benchmarking models on single graphs in transductive settings, but real applications require generalization to unseen graphs with different properties.
Experimental Design. We systematically vary three fundamental graph properties across families of 1000 graphs each: homophily range (00.0, 0.1], [0.4, 0.6), [0.9, 1.0)), average degree range (11, 5), [5, 10). [10, 20)), and cluster variance (basically feature noise, setting as options 0.2, 0.5 and 0.8), keeping all other graph family parameters at default values. We directly compare these inductive
'Aceuracy is appropriate given uniform community size distributions enforced across all graph families.

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Non-Mesaage- Passing GooteLLF
Message- Passing
Topelopical
Graph
Inductive Transsuctive
loomorphiom
A) Inductive
Transductive Performance
Average Degres
Homophily
Variance
Transductive
38
3/9
Degree
llcrophlly Value
Cludter Value
Intuctive
15:101
124,200
(6
(AAC
Pug Degres Range
llemophlyy Range
Clatter Vir ance Value
Distribution Shift Analysis (Inductive, Inference Only)
Positive Shift (+4.00)
Positive Shift (+ 0.30)
Positive Shift (+0.20)
Distribution
[1.5]
15.10]
[10.20)
[0.0,03]
10.4.0.0 61
[0...10)
0.8
Degroe fonge trrining)
Homoshily Rango (training)
Volue (craining)
Negative (-4.00)
Negative Shift (-0.30)
Negative ve (-0.20)
BiA
NW
[L.5]
[10,20]
[9.0.0.3] [0.4.0.6]
[0.0.100]
Figure 3: A) Inductive (graph families of 1000 graphs) versus transductive (single graphs) test accuracy community detection across different graph properties, with each architecture indi- vidually optimized. B) Distribution shift analysis: best-performing inductive models evaluated on graph families with shifted properties from the same Universe. Plots show accuracy changes under distributional shifts, with x-axis indicating the original training domain. N/A indicates shifts beyond feasible parameter bounds.
results against equivalent single-graph transductive evaluation using Graph World-style generation (with mean property of corresponding inductive setting). Figure3]4 reports the results. Appendix D extends the homophily analysis to specific heterophilic GNN architectures.
Insight 1: Distinct Model Ranking Profiles Across Settings. We observe striking divergence in model performance rankings between the inductive and transductive settings (Fig. 3 A). While GPS and non-message passing architectures (Deepset, GraphMLP) consistently achieve top and bottom performances, respectively, other architectures show clear setting- dependent strengths. For example, Neural Sheaf Diffusion excels inductively but falters transductively, suggesting its topological biases aid generalization across graphs. Conversely, GIN dominates transductively but fails inductively, indicating its success may stem from memorizing single graph's 's structure. T'hese shifts reveal that transductive performance is not always reliable proxy for model's ability generalize.
Insight 2: Transductive Setting Amplifies Graph Property Effects While increasing graph homophily and average degree improves performance both settings, these benefits are significantly amplified in the transductive paradigm. As seen in Figure 3 A. the performance gap between low and medium homophily or degree configurations is far more pronounced transductively. This suggests that when models have access to the entire graph structure during training, they better exploit favorable properties. Consequently, transductive evaluation may overestimate a model's 's true sensitivity to these structural characteristics.
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Basellne Performance
Distribution Shift Effects
Deepšet
nodes
GraphMLP
Community (1)
NAE Accurscy
1..
MAKE ARR Accuaayy
0.03
0,72
GCN
GraphSAGE
GIN
TopoTune
Triangle
NSD
Figure 4: Left:
baseline accuracy original graphs. Right: performance changes (A) when
evaluating on larger graphs (+200, +500 nodes). Triangle counting uses normalized MAE by average graph size N. Out-of-memory error for NSD in largest graphs (+500).
5.3 RQ2: How ROBUST ARE GRAPH LEARNING MODELS UNDER DISTRIBUTION SHIFTS?
Deployed models must handle property distribution shifts between training and test data, where target graphs exhibit different structural properties than those seen during training. Understanding how performance degrades under controlled property shifts is crucial for graph foundation model development.
Motivation.
Experimental Design. Using optimal model configurations identified in the inductive experiments of RQ1, we evaluate performance degradation under controlled shifts. That is, for each baseline property setting (homophily, average degree, cluster variance). generate, using the same Universe, a new test family with systematic shifts: +00.1 homophily, #44 average degree, and +-200 nodes per graph. The optimal models from RQI are evaluated the shifted families. This provides systematic characterization of each architecture's sensitivity to different types of distribution shifts.
Key Insight: Model Robustness is Context-Dependent, Not Universal. Our experiments (Fig. B) reveal that model robustness not an intrinsic property but emerges from specific interactions between model's ' architecture and the graph's 'ss properties. We found that identical distributional shifts can produce opposite effects depending on the training regime; for example, increasing homophily can harm a model's performance in a oowhooophhly setting but improve it in a medium one. This exposes prchitecture-specife degree graphs, while models like GraphSAGE show counterintuitive performance drop when homophily is increased from a low baseline. These findings suggest that models often achieve high performance through narrow specialization training regimes rather than robust generalization, highlighting the critical need for training data diversity.
vulnerabilities: GIN proves highly sensitive to degree shifts in low-
5.4 RQ3: Do MODELS TRAINED ON SMALL GRAPHS GENERALIZE TO BIGGER GRAPHS?
Motivation. Real-world deployment often requires models trained on smaller graphs to handle larger instances. Understanding size generalization is critical for practical scalability of graph learning models. While we focused until now on node-level community detection, we also evaluate triangle counting as a structural graph-leve task, though our framework supports other graph-level tasks.
Experimental Design. We evaluate generalization across graph sizes using two complementary tasks. For community detection (node-level, local task), we train on graphs ranging from 50 200 nodes, then evaluate on families (from the same Universe) with 250-400 and 550-700 nodes. For triangle counting (graph-level, global task), we follow the same size progression.
Key Insight: Graph-level MPNNs Fail to Generalize to Larger Graphs. We can see (Fig. 4 that node-level tasks (community detection) show minimal sensitivity to graph size (2% degradation) due to local neighborhood aggregation, except for GPS and NSD which suffer minor drops from their more global components (positional encodings, attention). Graph-level tasks (triangle counting) initially show only GPS, NSD, and GIN effectively solving the task, but while GPS and NSD maintain performance when scaling to larger graphs, GIN fails to generalize, suggesting traditional MPNNs overfit to training graph sizes.

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Grephilnivenn Eqieaalert
ComptrAbard Expatadert
Raw Performance Correlation
Rank Correlation
1.0
6:08
Spearien Cor atonn
6.0
Models
88
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Figure 5: Model ranking correlations between real datasets and equivalent synthetic datasets. Rank- ings computed via bootstrap analysis. GraphUniverse (blue) shows consistently higher aligmment with real-world model rankings compared to Graph World (purple) across both raw performance and rank-based metrics. "Without Baselines" excludes DeepSet and GraphMLP to avoid overestimation.
5.5 RQ4: DOES GRAPHUNIVERS ACCURATELY PREDICT REAL-WWORLD MODEL PERFORMANCE?
Motivation. A critical question for any synthetic benchmark is whether insights gained extend to real-world performance. We demonstrate that GraphUniverse effectively predicts model behavior on real, inductive datasets, providing stronger alignment than single-grapl approaches.
Experimental Design. For five real-word wollwwrld inductive datasets we extract key structural properties and generate two synthetic equivalents: (1) a Graph equivalent matching property distributions (5th-95th percentiles), and (2) Graph World equivalent using mean values in single graph. We map dataset-specific communities to enable fair comparison. We train models on original tasks for real-world datasets and community detection for synthetic datasets. After optimizing the same suite of models as in previous RQs, we compute model rankings via bootstrap analysis and assess correlation between mean synthetic and real rankings. Full extraction details in Appendix K.
Givereniverse Provides Superior Real- World Alignment. Figure 5 shows GraphU- niverse achieves substantially higher correlations with real datasets than Graph World. We analyze both all models and graph-aware models separately, as message-passinng baselines (DeepSet, GraphMLPP consistently underperform across all settings, artificially inflating correlations. Focusing on ghessage-passing models reveals that Graph shows positive correlations for all datasets while Graph World shows negative correlations for half. Model-level analysis in Figures and [] of Appendix
Key Insight:
K.2 confirms that GraphUniverss accurately captures how model rankings vary across datasets with different structural properties. These results validate Graph Universe as a meaningful proxy for real-world evaluation, particularly for rapid prototyping and systematic studies.
CONCLUDING REMARKS
We introduce Gnipeeniverse, a synthetic graph generation framework designed to address a critical gap graph learning: the systematic evaluation of inductive generalization. By generating graph families with consistent semantics and tunable structural and feature properties, GraphUniverse provides the scalable, controlled data required to rigorously assess a model's ability to generalize across diverse and unseen graphs. Our experiments reveal valuable insights into model robustness and generalization. We release GraphUniverse as an open-source tool to unlock new research directions in the principled development and validation of both existing and novel architectures -including potential applications in graph foundation model research, though such extensions would require additional development beyond current framework.
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REPRODUCIBILITY STATEMENT
GraphUniverse is available as a Python package at htttps://yyp..org/ wnnojeettgraap--nieeeeeee.e with full source code at https://github.com
All experiments can be reproduced using
the TopoBench framework framework al. (2025)
ACKNOWLEDGMENTS
This work was supported by Grant PCI2023-1459742 funded by MI- TW//EE/11.13099/// 10001 1033 and cofunded by the European Union (GRAPHS4SEC project). This work is also supported by the Catalan Institution for Research and Advanced Studies (ICREA Academia). Louis Van Langendonck is supported by the predoctoral program AGAUR-P ajuts (2025 FI-3 00065) Joan Oró, of the Department of Research and Universities of the Generalitat of Catalonia, as well as the European Social Plus Fund. Nina Miolane acknowledges the NSF grant 2313150 and the NSF CAREER Award 240158. Guillermo Bernardez Gil acknowledges funding from Chan Zuckerberg Initiative, UC Noyce Foundation, and Arlequin Al.
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A USE OF LARGE LANGUAGE MODELS (LLMS) IN PAPER WRITING
We drafted all content ourselves and then used LLMs to improve grammar, rephrase text, and shorten extensive sections. The models are used as editorial tools to help make our writing clearer and more concise.
B DISCUSSION ON ROLE AND POTENTIAL IMPACT IN GRAPH FOUNDATION MODEL DEVELOPMENT
While our primary contribution focuses on systematic inductive evaluation, GraphUniverse has potential relevance to graph foundation model (GFM) development. This section outlines two potential applications.
B.I SYSTEMATIC EVALUATION TESTBED
GraphUniverse enables rapid generation of diverse graph families for comprehensive GFM assessment across controlled distribution shifts. Unlike static benchmarks, our framework allows systematic robustness evaluation by generating unlimited scenarios with known variations in structural properties (homophily, density, degree distributions). This provides a precise, low-cost method for identifying architectural vulnerabilities and understanding failure modes, directly addressing concerns about current evaluation practices raised recent position papers.
B.2 DATA AUGMENTATION POTENTIAL.
Our framework could potentially be extended to serve as a sophisticated data augmentation tool for GFM pre-training. By fitting GraphUniverse parameters to match statistical properties of target domains, researchers could generate novel but realistic graph instances that preserve essential structural relationships. This aligns with successful synthetic data strategies in computer vision and NLP, where augmentation has proven valuable for improving model robustness Tobin et al.) (2017); Wang al. (203).
The key strength this approach would be leveraging our framework's 's flexibility to generate datasets covering data modalities completely unseen in original training sets. By systematically varying homophily levels, density, and graph sizes beyond those present in real datasets, researchers could potentially reduce overfitting to specific dataset characteristics and train more general models capable of robust performance across diverse graph types.
B.3 LIMITATIONS AND FUTURE DIRECTIONS
These applications represent potential future directions rather than validated capabilities, with several concrete challenges requiring resolution. First, identifying meaningful "communities" -th building blocks of our framework-in real-worls datasets presents domain-specific challenges. While some datasets naturally provide community structure (e-g., atom types in molecular data), others would require sophisticated clustering approaches, potentially incorporating positional information or employing a two-stage process: initial unlabeled pre-training followed by clustering on learned embeddings to fit the data generator.
Second, realistic feature generation would require careful fitting to real data distributions, most straightforwardly achieved by computing per-community, per-dimension statistics (means and stan- dard deviations) and sampling accordingly. Our current DC-SBM foundation may require extension to more sophisticated generators to fully capture the complexity needed for realistic augmentation across diverse domains.
Nevertheless, recent GFM surveys explicitly call for these types of evaluation and generation capabil- ities, suggesting clear alignment with community needs and representing a promising line future work Wang et al. 2025): (Bechler-Speicher al. 2025
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C BERNOULLLI FORMULATION OF SIMPLE DC-SBMS
Poisson multigraph. In the classical DC-SBM (Karrer & Newman 2011 edges are Poisson counts
Ag ~ Poisson(Ais), with per-community normalization Eier 0, = 1. The expected block edge totals are then
- -,,0A,,,b,
(4)
[Arai (
r 8, s.
E[Mrs]
(5)
Collapsed Poisson simple graph. If we form simple graph by collapsing multi-edges,
Āι { (Ai 2 1},
(6)
then
Pr(A,, = 1
(7)
Since 1 for O, the collapsed model systematically underestimates edge probabilities and thus block totals, except in the extremely sparse regime where - Aij z Xsj-
Bernoulli simple graph. Following the approach Rohe al. 2018), we can define edges directly as Bernoulii trials,
Aij ~ Bernoulli (min [1,0,0,Po.M. Po.b,))
(8)
with per-community mean-one normalization
Σ V
(9)
The expected block edge totals are then:
For r + [Msss] Pre Eier Ejes Nae,000, 0,0,
For B[Mr.] Pr. Eier Ejen,ys 0:0; (excluding self-loops)
.
Under the normalization constraint, these simplify to:
For r * 8: Mrs] = Pr.IVV For r = 8: [ P. VV.0V
Equivalence.
To match the Poisson multigraph block expectations, we set:
Ars
Aur
Prs
s),
Per
(r=s).
(10)
V
However, noted by Rohe al. 2018 this equivalence only holds when the resulting probabilities satisfy 0,0, Pr,8, < 1 for all j, Bernoulli probabilities:
When this constraint is violated, we apply clipping to ensure valid
edge probability = min(1, P,0,P..o,).
(11)
which introduces a systematic deviation from the Poisson block structure in dense regimes.
Theoretical justification. The theoretical foundation for this approach follows directly from Theorem 3 in Rohe et al. 2018). Their result shows that in sparse regimes where Xij = O(an/n) for some sequence a,, there exists coupling between the thresholded Poisson graph and the direct Bernouli graph such that
E ((A)
BII BI
O(an/n),
(12)
where t(A) represents the thresholded Poisson graph and B represents the direct Bernoulli graph. This establishes that the approaches asymptotically equivalent in sparse settings.
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Summary. The Bernoulli DC-SBM with clipping preserves the interpretability of edge probabilities and avoids the systematic underestimation of the collapsed Poisson approach, while providing exact control over the simple graph structure. The theoretical equivalence established by Rohe al. (2018) validates this approach in sparse regimes, while the controlled deviation from Poisson block expectations due clipping represents principled trade-off that enables direct generation of simple graphs with desired structural properties.
D DISCUSSION ON THE LIMITATIONS OF DEGREE-CORRECTE STOCHASTIC BLOCK MODELS AS DATA GENERATOR
The DC-SBM formulation underlying GraphUniverse carries inherent limitations that merit explicit discussion. We categorize these limitations into types, each with different implications for framework's applicabiiity and future development.
D.1 READILY EXTENSIBLE LIMITATIONS
Several limitations stem from design choices made for experimental simplicity and interpretability. Features such as deterministic community membership, discrete non-overlapping communities, and uniform community size distributions could readily be implemented as extensions to our current framework. We deliberately chose these simplifications maintain clear experimental control and focus on core structural phenomena that are easily identifiable, controllable, and translatable to real-world settings.
For researchers interested in studying specific phenomena like overlapping communities, gradual community transitions, or hierarchical community structures, our framework provides solid founda- tion that can be extended while preserving the systematic control that makes synthetic benchmarks scientifically valuable.
D.1.1 IMPLEMENTING OVERLAPPING COMMUNITIES
To illustrate the extensibility of our framework, we outline how overlapping communities could be implemented through two additional universe-level parameters:
Co-occurrence Count Distributions: For each community k, define discrete probability distribu- tion over the number of additional communities node can belong to. This could be implemented as user-defined distributions or generated using negative binomial distributions with controllable parameters for mixing probability and distribution shape. Setting all distributions to concentrate on zero recovers the current non-overlapping scenario.
Community Mixing Matrix: A symmetric K x matrix M where each row sums to 1, controlling how membership distributed among overlapping communities. Entry M.,, represents the relative strength of membership in community j when node's primary assignment is to community i.
The extension process would work follows:
1. Assign each node primary community before
Sample the number of additional communities from co-occurrence distribution
3. Randomly select additional communities and use the mixing matrix to determine member- ship weights
4. Compute final node behavior as weighted combinations edge probabilities become weighted by both nodes' membership vectors, degree factors determined by the dominant community membership, and features are drawn as weighted combinations from respective community centroids
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Formally, if node i has membership vector πT; E [0, 1|K with 2k=1 Tel 1, then:
Pij min 1,0,0,
Σ
Tir Tir Tsaaell s]
(13)
0, Os where K- arg max
(14)
K x4 ~ N
Σ
To I²I
(15)
This approach preserves all existing framework properties while enabling systematic control over community overlap patterns, demonstrating how our hierarchical design facilitates principled exten- sions.
D.2 FUNDAMENTAL LIMITATIONS
More significant limitations arise from our inability to directly control complex motif-driven structures, geometric arrangements, or specific higher-order patterns commonly found in real-world networks. The DC-SBM cannot generate graphs with predetermined triangular motifs, star patterns, geometric constraints, representing a fundamental constraint on the types of graph structures our framework can produce.
This limitation could potentially bias our evaluation toward models that perform well on community- structured data while potentially penalizing architectures designed for other graph topologies. How- ever, our Research Question 4 (Section 5.5) experiments provide encouraging evidence that despite these structural constraints, GraphUniverse -eeeeerrreeeegeeerteeeddd datasets effectively predict real-world model performance across diverse tasks, including molecular property prediction that depends heavily on complex functional groups and higher-order chemical structures (Wu al.) 2023).
D.3 IMPLICATIONS AND FUTURE DIRECTIONS
The transfcrability observe suggests that community- centric evaluation captures sufficient funda- mental graph learning capabilities- tth interplay between local structure, features, and connectivity patterns for meaningful model assessment across diverse domains. Nevertheless, extending framework to incorporate more sophisticated generative models while maintaining systematic experi- mental control represents valuable direction for future work, particularly for applications requiring finer control over specific structural motifs or geometric properties.
E SCALING RAW PROPENSITY TO BERNOULLI PROBABILITY MATRIX WITH DESIRED EXPECTED HOMOPHILY AND AVERAGE DEGREE
To introduce controllable heterogeneity, generate propensity matrix P e kxk as
Pr. 1 + Ers: E.s ~ N (2 E [0, 1],
(16)
where e controls the variance of the perturbation. Entries are clipped to [o, 2] and symmetrized by setting P, = Par- When = 0 the matrix reduces to the all-ones matrix, while larger values of e yield increasing heterogeneity across communities.
Scaling to Target Density and Homophily The raw propensity matrix P only specifies relative propensities. Given graph with nodes with uniform distribution of k communities, want to transform it into valid probability matrix p E [0, xx that achieves a user-specified average degree d and homophily level h.
Let
Saing
P mr
Saft
Σ Pre.
r,s=1
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We first apply two scaling factors Cddiag Cof > 0 to obtain
adiag Per,
= 9,
Prs
(17)
Cfff Pres
r + s.
The ratio of diagonal to off-diagonal mass then
Σn Pi
Odiag Sdink
Er Pl Ooff Soff
We now enforce the target homophily constraint by stating enforcing that this ratio equals h/(1 -h). which yields the constraint
Oddaa
h
Soft
(18)
Afff
Salag
Up to this point we are not yet working with actual probabilities so we can scale the diagonal by setting Ording and calculating Ooff by solving equation 18 and scale the off-diagonal by this value.
Next we impose the average degree constraint and scale obtain actual edge probabilities. Let be the number of nodes and let the target edge density be
Ptarget
(19)
71 n-1
We apply a global scaling factor 3 > 0 to obtain the final matrix
P = 3P',
(20)
where B is chosen such that the mean entry of p equals Ptarget, i.c.
n² Ptarget
(21)
Σr, r, Pis
Finally, we clip entries of P* to the interval [0, 1] to ensure valid Bernuulii probabilities.
The resulting matrix P* satisfies three properties: (i) it preserves the relative heterogeneity induced by P. (ii) it achieves the specified homophily ratio between intra- and inter-community connections, and (ii) it yields expected average degree of d up to sampling fluctuations. This construction allows graphs to be generated at arbitrary density and homophily levels without discarding the fine-gradt mne-grained structure encoded in P. which is essential for controlled multi-graph family generation.
F DETAILS OF COMMUNITY-COUPLL DEGREE FACTORS
For completeness we record the precise definitions used in the Begree-commine coupling mecha- nism.
Overall procedure.
The coupling process consists of four steps: (1) sample power-law degree factors independently, (2) sort them in ascending order, (3) assign sorted factors to nodes based on community-specitte rank sampling, and (4) apply global normalization.
Power-law degree factor generation. We first generate n. independent degree factors from power-law distribution with exponent a:
PowerLaw(a), i= i,.,..,, These then sorted to obtain the ordered sequence (0(1), ... (())) with (11) I S 0())-
Community rank centers. Each community A: is assigned degree center d, E [-1, that maps linearly to a preferred mean rank:
1 1 dz (n 1n-1)
Thus & = 1 corresponds to rank μa = 0 dlwees-degree regime), 8 = 0 to rank f4N = (n 1)/2 (middle-degree regime), and dz +1 to rank μ4 (highest-degree regime).
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Rank sampling and assignment. For each node in community c(t), sample rank index L from a truncated Gaussian distribution:
4 ((a(() x²) truncated to [1, n²]
Node i is then assigned degree factor 8 ((,))
Variance interpolation.
The sampling variance set as
0² i + (1 - p) max min))
where P [0, 1] is the degree separation parameter. Here m corresponds to nearly uniform assignments with strong overlap across communities.
Minimal variance. space, we set
To prevent degenerate overlaps when communities are spread apart in degree
min max (L.
Pa-Merl
min
which ensures sufficient separation whenever the community centers μi are far apart.
Normalization. In the classical Bernoulli DC-SBM, degree factors are normalized within each community:
Σ 0:
=1
Vr.
fT b.
In our construction instead apply single global normalization
0 1-
so that the average degree factor is one across all nodes. This choice preserves relative placement of communities in the degree spectrum, though it does not guarantee exact per-community calibration. Empirically we observe that the global normalization suffices for maintaining the target average degree (see Section 4.3)
G CONNECTIVITY CORRECTION ALGORITHM
When the initial edge sampling results in disconnected components, employ a greedy algorithm to restore connectivity while minimally perturbing the intended block structure. The procedure operates as follows:
Algorithm Overview: We iteratively connect the smallest disconnected component to the main graph by selecting edges that best align with the target probability matrix Psub- After each edge addition, we recompute connected components and repeat until the graph becomes fully connected.
Connection Selection: For each potential edge (i.j) between communities e(i) and c()), calculate score based on the current deviation between actual and expected inter-community edge probabilities:
If the actual probability is below the expected value (actual < P Puub (el)) c())
adding edge reduces this negative deviation (preferred option).
If the actual probability exceeds expectations, we select connections that minimize further deviation.
Deviation Calculation: We maintain normalized actual probability matrix where edge counts divided by the maximum possible edges between community pairs, then scaled to match the total mass of Subb for fair comparison.
This approach optimally balances connectivity requirements with structural fidelity: it improves the match to the target block structure when possible (by connecting mnder-coneected community pairs) and minimizes degradation when connectivity necessitates violating the intended structure, thereby preserving the statistical properties of the generated graph family the greatest extent possible.
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H CORE PARAMETER ALLOWED SAMPLING RANGES FOR VALIDATION EXPERIMENT
Table 2: Parameter sampling ranges for randomized validation experiments
Parameter
Type
Sampling Range
Universe Level
Edge Propensity Variance (c)
Continuous
[0.0. 1.0]
Feature Dimension
Discrete
[10, 100]
Center Variance (o
ognter)
Continuous
[0.1, 1.0]
Cluster Variance (o cluster)
Continuous
[0.1, 1.0]
Family Level
Min Node Count (Amin)
Discrete
[50, 400] [100, 1000]
Max Node Count (7max
Discrete
Min Communities (kmin)
Discrete
[2. 15]
Max Communities (kmax)
Discrete
[4, 15]
Homophily Range (huin Mmaxx)
Range
[0.0, 1.0]
Average Degree Range (cimnn dmaxx
Range
[2.0, 20.0]
Degree Separation Range (Pmin Pmaxx)
Range
[0.0, 1.0]
Power Law Exponent Range (Ommin Omaxx)
Range
4.5]
For parameters that represent ranges themselves (e.g., Homophily Range, Average Degree Range), we sample the range bounds from the specified limits and then generate individual range spans with widths between 5% and 20% (randomly drawn) of the parameter space, ensuring meaningful variation while maintaining practical constraints. All experiments use fixed universe size of K 15 communities. Results shown in T
Note: For paired parameters (min/max node count and communities), the code ensures logical constraints where maximum values exceed minimum values by appropriate margins.
1 VALIDATION METRICS IMPLEMENTATION DETAILS
We organize our validation metrics into three categories that capture different aspects of generation quality. An overview table of all validation metries is given in Table
1.1 GRAPH PROPERTY METRICS
These metrics verify that generated graphs match their target structural specifications:
Homophily: Fraction of edges within communities: h Enner [[((i) c(j)\/1B). This directly validates whether the target homophily level is achieved.
Average Degree: Mean node degree across the graph, validating that edge density scaling produces the intended connectivity level.
Degree Tail Ratio: Ratio of 99th percentile to mean degree (79g dgg/d), capturing heavy-tailedness of the degree distribution and validating power-law parameter effects.
Generation Time: Wall-clock time per graph instance, assessing computational efficiency.
Mean Probability Matrix Deviation: This metric quantifies how well the realized graph structure matches the target community connection patterns. For each graph, we compute the deviation between the actual probability matrix Aqutual and the expected matrix P subi
1. Calculate actual edge probabilities between communities i and j:
edge.count,
if =
Aacuul [i
( 1) edge_count_ of n, R2
(22)
if +
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where is the size of community i and edge_count,, is the number of edges between communities i and
2. Compute mean absolute deviation: deviation + Eis [8, P Suub Puub(i;)) [i:
1.2 SIGNAL STRENGTHI METRICS
These metrics assess whether within each graph can find meaningful, learnale community structure through different node-level predictive signals:
All signal metrics Random Forest classification with the following configuration: 100 estimators, unlimited depth, minimum 2 samples per split, minimum sample per leaf. Data is split 70/30 train/test with stratification ensure all communities appear both sets. Performance is measured using macro Fl-score.
Feature Signal: Uses node features x, directly as input to predict community labels.
Degree Signal: Uses node degree d, as single-dimensional input.
Structure Signal: For each node v, construct feature vector f, E TR by concatenating community
neighbor counts at distances 1, 2, and 3:
(1) n. (3)) (d) [m p2.2 m. and na,e is the number of neighbors of node in commu- nity at exactly distance d. For example, with 5 communities, if node has neighbors in com- munities [1,1,2,4.4.4] at distance 1, [2,5,, at distance and [1,4,4] at distance 3, then [2, 1, 0, O, 1,000,,,
(2)
where n. (d) (d)
(d)
(d)
1.2.1 CROSS-GRAPH CONSISTENCY METRICS
These metrics evaluate whether community identities remain semantically consistent across different graph instances within a family:
Structure Consistency: For each graph g. compute
consistency, g PSpeamaan (P (Pi., A (9) s.
where PSpearman denotes Spearman rank correlation, Pt.. is row of the universe propensity matrix restricted to participating communities, and A (g) actual, the corresponding row of the actual probability matrix.
Degree Consistency: Combines within-graphe and cross -uass-graphhh consistency:
consistency, 2
Piithin +
Prross
(g)
where Pwithin PSpearman (9) deton) degree centers 6c and
compares average degrees per community d(s) with universe
Σ Wa.n PSpearman (s(0), s(o')
Peross
Eg'to Wg.g
9'#9
where s(s) E [0, 1 the percentile signature for graph g with sc
(x) rank (mnk(d!s))
[C00] for participating
communities and (( NaN otherwise, Wg.g' [c (g) NaN A (g')
NaN) is the overlap
weight, and the correlation is computed only over non-NaN entries.
Feature Consistency: Average pairwise cosine similarity between community centroids:
N-1
k
2
Σ
consistency
N(N
s=1
g'=91
where N is the number of graphs and (1( is the centroid of community e in graph g.
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Category
Metric
Description
Homophily
Fraction of edges within communities: h = E EeSEE [c((i) =
Graph Property
Average degree
Mcan node degree, validating edge density scaling
Degree tail ratio 99
Ratio of 99th pereentile to mean degree: Tap dge/
Generation time
Wall-clock time per graph instance
Mean Probability Ma- trix Deviation
Average deviation between realized and target community edge probability matrices (Pubb)
Feature signal
Per-graph, node-level community predictability via Random Forest (macro F1) using node features x, as predictor
Signal Strength
Structure signal
Per-graph, node-level community predictability via Random Forest (macro F1) using k-hop neighbors label counts (k E (1,2,3) as predictor.
Degree signal
Per-graph, node-level Community predictability via Random Forest (macro F1) using node degree d, as predictor.
Feature consistency
Average pairwise cosine similarity between community feature centroids across graphs
Cross-Graph Consistency
Structure consistency
Spearman correlation between universe propensity matrix P and realized edge probabilities
Degree consistency
Rank correlation of community degree orderings across graphs
Table 3: Validation metrics for evaluating GraphUniverse generation framework.
J EXPANDED VALIDATION RESULT ANALYSIS
Detailed Validation Result Analysis. The correlation heatmap (Figure demonstrates compre- hensive parameter control across all validation metrics, revealing both expected relationships and theoretically interpretable effects.
Graph Property Metries (first panel) show expected strong correlations with some additional insights. Input parameters precisely control observed homophily and average degree, while power-law exponent governs degree tail heaviness and node count correlates with generation time. We observe minor statistically significant effects on generation time from other parameters, likely reflecting computational complexity variations. Notably, increasing node count reduces probability matrix deviation, suggesting that larger graphs experience fewer random sampling effects due to improved statistical power. Slight deviations from target edge probability matrices under high average degree and degree separation parameters reflect the multiplicative edge generation process, where stronger degree factor effects naturally influence connection patterns.
Signal Strength Metrics (second panel) demonstrate strict adherence theoretical expectations the single-graph level, where Random Forest classifiers predict community labels within each graph instance. Cluster variance directly controls feature signal strength by determining feature separability between communities. The negative correlation between community count and all signal metrics reflects the fundamental difficulty of multi-class classification: distinguishing between two commu- nities is inherently easier than discriminating among many, leading to higher Fl scores with fewer positii classes given similar discriminative power. Homophily's 's positive correlation with structure signal has clear mechanistic ex explanation: when neighbors predominantly share the same community label, neighborhood composition becomes a highly predictive feature for node classification. This relation- ship thip-where averaging neighborhood representations provides strong community signals-likel) explains the effectiveness of simple GNNs like GCN homophilic settings. Similarly, average degree enhances structure signal by providing more neighborhood information, giving classifiers richer structural context for community prediction.
Cross-Graph Consistency Metrics (third panel) provide the strongest validation of our hierar- chical design. The intended universe-level signals are present and tightly controllable; propensity variance governs structure consistency, degree separation controls degree consistency, and cluster
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variance determines feature consistency. The negative correlation between community count and consistency metrics reflects increased sensitivity to random effects when computing correlations across many classes, where small sampling variations more easily perturb rank orderings. We observe a theoretically justified trade-off between degree separation and propensity variance effects on edge probability deviation, emerging from our multiplicative edge generation process where Po = 0,0, scate((((() c()], causing these parameters to modulate different components of the same generative mechanism.
PARAMETER-VALIDATION J.1 INDIVIDUAL PARAMETER- VALIDATION PLOTS
Please see figures below.
Enge Propersity
Claster
Emper
Dag
000
(a) Randomized Parameter Validation of Edge Propen- sity Variance parameter. Top left shows Pearson cor- relation and statistical significance level (NS, not sta- tistically significant).
(b) Randomized Parameter Validation of Cluster Vari- ance parameter. Top left shows Pearson correlation and statistical significance level (NS, not statistically significant).
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Noon Hanoopicyy
Wean Asurage Degins
(a) Randomized Parameter Validation of Homophily Range parameter. Top left shows Pearson correlation and statistical significance level (NS, not statistically significant).
(b) Randomized Parameter Validation of Average De- gree Range parameter. Top left shows Pearson corre- lation and statistical significance level (NS, not statis- tically significant).
Moon Cormeraniion Partccpating
Pestuee Signal
Dayon of
Fesum
Southan Eignal
Digies Co
(a) Randomized Parameter Validation of Node Count range parameter. Top left shows Pearson correlation and statistical significance level (NS, not statistically significant).
(b) Randomized Parameter Validation of Communities Participating Per Graph Range parameter. Top left shows Pearson correlation and statistical significance level (NS, not statistically significant).
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Vear Degres Separstion
Maan Plawer Law Expasent
Grogh
Uapa Prcko
(a) Randomized Parameter Validation of Degree Sepa- ration Range parameter. Top left shows Pearson corre- lation and statistical significance level (NS, not statis- tically significant).
(b) Randomized Parameter Validation of Power Law Range parameter. Top left shows Pearson correlation and statistical significance level (NS, not statistically significant).
K REAL- DATASET VALIDATION
K.1
EQUIVALENT DATASET PARAMETER EXTRACTION
To establish correspondence between real-world datasets and their synthetic equivalents, we developed a systematic parameter extraction pipeline that maps dataset characteristics to GraphUniverse and GraphWorld generation parameters.
K.1.1 COMMUNITY DEFINITION STRATEGY
The extraction process begins by identifying suitable notion of "community" for each dataset, which varies based on available features:
Molecular datasets (NCH1 MUTAG, AQSOL, OGBG-MolHIV): We use atom types as natural communities, extracted via argmax on one-hot encoded features or directly from atomic number features.
IMDB-MULTI:
Lacking node features or labels, we perform degree-based clustering using K-means on node degree and clustering coefficient features, with optimal K determined via silhouette scoring.
K. PARAMETER MAPPING METHODOLOGY
For each dataset, we extract:
1. Graph size distribution:
Number of nodes across all graphs
2. Average degree distribution: Mean degree per graph
3. Community structure: Number of unique communities per graph and total unique commu- nities
4. Homophily: Fraction of edges connecting nodes in the same community
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For GraphUniverse use the 5th-95th percentile range of each property capture the full distribution while avoiding outliers. The universe size K is set to the maximum of (i) the 90th percentile of communities needed to cover 90% of nodes, and (ii) the maximum communities per graph, ensuring sufficient diversity.
For Graph World, following their single-graph paradigm, K' and communities per graph are set to the dataset's 's mean unique communities per graph. However, for fairness, we set the graph size to 1000 instead of the average of the real dataset's graph sizes, which in general are too small in an inductive dataset to train a model on.
mean values for all properties. Both
K.1.3 EXTRACTED PARAMETERS
Table presents the extracted parameters for all datasets. Note that GraphUniverse captures the heterogeneity of real datasets through ranges, while GraphWorld reduces this to point estimates.
Table 4: Extracted parameters from real datasets and their synthetic equivalents. GraphUniverse uses 5th-95th percentile ranges; Graph World uses mean values.
Nodes
Avg. Degree
Homophily GU Range
Communities
Universe
#Graphs
Range
GW
Range
GW
Range
GW
OGBG MolHIV
41,127
[13, 46]
1000
[2.00, 2.50]
2.14
[0.32, 0.84]
0.61
5]
5/3
AQSOL
9,833
[10, 36]
1000
[1.60, 2.25]
1.98
[0.15, 0.92]
0.59
[2, S]
5/2
IMDB MULTI
[10, 31]
[4.67, 17.00]
[0.35,
0.80
[2.5]
NCTI
4,110
[15, 59]
1000
[2.00. 2.50]
2.16
[0.38, 0.82]
0.62
[]
3
5/3
ZINC
10,000
[16. 31] 1000
[2.00. 2.50]
2.14
[0.32. 0.71]
0.52
[]
Notes: GU GraphUniverse, GW Graph For Universe K. we show both values (GraphU- niverse/GraphWordd when they differ. For all experiments except the IMDB-MUTE one, other generation parameters (edge propensity variance, cluster variance, degree separation) are set to mid-range values (0.5) for consistency across experiments. For the not have any node-features, we set the feature signal to zero in equivalent dataset (center variance of 0.01, cluster variance of 1.0) and the degree separation, now being the main identifier for community detection, to a range of 0.9 to 1.0.
one, since it does
K.2 DETAILED REAL- WORLD ALIGNMENT ANALYSIS
Figure 10 provides a detailed model-level analysis of ranking correlations between synthetic and real-world datasets. For each of the six datasets, we plot individual model rankings computed via bootstrap analysis (1000 iterations) quantify uncertainty in rank estimates. In this plot we omitted the baseline models, since these artificially inflate correlation. This effect is displayed in Figure 11 where the DeepSet and GraphMLP models are included.
The results confirm that GraphUniverse preserves real-world model ranking patterns across diverse datasets. GraphUniverse (top row) achieves strong positive Spearman correlations, demonstrating that models maintaining similar relative performance in both synthetic and real settings. In contrast, GraphWorle (bottom row) shows poor alignment with multiple negative correlations, failing to capture how model performance varies across different graph structures. This ranking preservation is crucial for practitioners who need to select architectures based on synthetic benchmark results, as Graph Universe reliably predicts which models will perform well on real-world tasks.
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Modi
Spearnan Rank Porreations
OGRG-MelIIY
ADSOL
IMOD-WFULTI
ZINC
ANG
2--4108--766
0--d0pp--- e00))
1- 500g -0499)
Greetttant
Grenaue
Figure 10: Model ranking correlations between real and synthetic datasets. Each point represents model's rank (1=best) with eπror bars showing bootstrap standard deviation (1000 iterations). GraphUniverse (top row) demonstrates strong rank preservation, while Graph World (bottom row) shows poor alignment including negative correlations. The red line shows linear fit to mean ranks, with the diagonal gray line indicating perfect correlation. Non-message passing models (DeepSet and GraphML P)) omitted focus graph-aware architectures, they consistently underperform and artificially inflate correlations.
Spasmman Rank Commlatione MC11 2- 0-0.000 2 0(NG)
Ad60L 677 g- DARS
2NC 6210 0.6G)
0.777 8.97.
2. 2.03 5.265
Deng
Greethonn
Spd Sawpe
Rox Dataes
Tee Celted Purk
Nad Debari Tark
The Ooand Purk
Rod Nalo
Figure 11: Model ranking correlations between real and synthetic datasets. Each point represents model's rank (I=best) with error bars showing bootstrap standard deviation (1000 iterations). The red line shows linear fit to mean ranks, with the diagonal gray line indicating perfect correlation. Non- message passing models (DeepSet and GraphMLP) are incleded, artificially inflating correlations. especially for the GraphWorld case (bottom row).
L HETEROPPHLLY-SECCIIIII ARCHITECTURES EVALUATION
To further validate our framework's ability to capture nuanced architectural differences across graph properties, we extend our benchmarking to include models that have shown effectiveness in heterophilic settings: Frequency Adaptive Graph Convolutional Network (FAGCN) (Bo al. 2021). H2GCN (Zhu et al. 2020), and ChebNet (Tang et al. 2024). While FAGCN and H2GCN explicitly designed for heterophilic graphs, ChebNet's spectral approach using Chebyshev polynomials has demonstrated strong empirical performance in low-homophily settings, making it valuable addition our analysis.
L.1 EXPERIMENTAL SETUP
We evaluate FAGCN, H2GCN, and ChebNet alongside our original model suite across five homophily levels (0.05, 0.25, 0.5, 0.75, 0.95) in the transductive setting (GraphWorld style) and three homophily
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ranges ([0.0,0.1], [0.4,0.6], [0.9,1.0]) in the inductive setting (GraphUniverse style). All other experimental parameters remain consistent with our main benchmarking protocol (Section 5.1p. including hyperparameter optimization procedures and evaluation metrics.
The hyperparameter grid used optimization these heterophily-speciilize models is displayed in Table 5
L.2 RESULTS AND ANALYSIS
Figure 12 presents the performance of peterophily-specialiiie models compared to our baseline architectures across the homophily spectrum.
Our results reveal surprising differences between evaluation paradigms:
Transductive Performance: As expected, H2GCN and ChebNet demonstrate superior performance in the most heterophilic regimes (0.05-0.25), validating their design principles for heterophilic graphs. Their advantage gradually diminishes homophily increases, with performance converging to baseline levels at high homophily (0.95). Notably, FAGCN underperforms relative to expectations, suggesting that its frequency-adaptive mechanisms may translate effectively to our synthetic community detection task.
In contrast, none of the heterophily-specciane advantages in the inductive setting. Performance differences across homophily levels become less pronounced, and specialized architectures show no clear superiority over standard message-passing networks like GAT and GraphSAGE. This finding suggests that the benefits of esterophily-speciine designs may diminished when models must generalize entirely new graph instances rather than leveraging global structural patterns within single graph.
Inductive Performance:
models maintain their
This experiment further validates GraphUniverse's capacity to reveal architectural behaviors that remain hidden in traditional single-graph evaluations. Architectural advantages observed in trans- ductive settings may not transfer to inductive scenarios, emphasizing the importance of evaluation paradigm choice.
Table Hyperparameter grid search space for heterophily-sspeciaiiee models.
Model
Hyperparameter
Search Space (Grid)
ChebNet
out wwt.loottoooooo... chhannels
(3)
www.uurreeneddr.e.. dropout
(0.3)
backbone, layers
(2.4)
backbone..t
(2
backbane. .ooral izztii
(sym, rw)
backbone ddopout
0.44
beadout.hidde badde readout.
[[16). II]
(0.3)
FAGCN
Festure_eneader.. http:/ccccdeer.tttcccceeeee ott_nntntcceesssseeaaaaae
[32, 64)
reature encoder, proj dropout
(0.3)
rww.booeerru.layeers
(2.4)
backbone.epo
(0.0.0.1.0.2)
backbone.. saackbone.rormaaie. normalize
Falsee
backbane, drrppattedrropaut
(.22.
layers readout.dropout
[166).
[0.3]
H2GCN
feature encoder. out channels
(32,64)
(0.3)
backbone saakbbnee.umm layers
(2.4)
backbone.
[3
backbane. use_relu
True, False)
backbone dropout
(0.2.
readcut. beaucuut.hideeee layerr
{116), [)
(0.3)
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Non- Mese-Messaggeeee Passing
Messag e-Passing
Seaterophlly-Specialieeed Specialized
Topological Shawl Demation (MSD)
Graph Transformers Twralomar 10751
CespSet
Carnelutiou
Prequancy Adestive GCN 9902)) ChesNict
GraptML
Grapn Aitennio ketwork (GUM)
Merwork
0.8
Accuracy
0.6
0.5
[0.
0.1]
[0.4, 0.6]
[0.9, 1]
Homophily Range
Transductive
Accuraty
0.6
0.4
0.25
0.5
0.75
0.95
Homophily
models (FAGCN, H2GCN, ChebNet) Figure 12: Performance comparison of homophily levels in inductive and transductive settings. against baseline architectures across varying Error bars represent the standard deviation of the test performance across different random seeds.
M FURTHER EXPERIMENTAL DETAILS
This appendix provides supplementary details regarding the experimental setup used in this work to ensure reproducibility.
M. GRAPH GENERATION PARAMETERS
Table/@|spccifies the default generation parameters used for both the primary inductive setting and the baseline transductive setting. The universe parameters define the underlying semantic space (e.g., number communities, feature characteristics), while the family parameters control the structural properties of the sampled graphs. Note the key differences: the inductive setting generates large family of smaller, varied graphs, whereas the transductive setting generates a single, large graph with fixed properties.
M.2 BENCHMARKED MODEL ARCHITECTURES
This section provides brief overview of the models included in our experimental evaluation.
GraphMLP DeepSet These models serve as non-message-pasinn baselines. DeepSet is permutation-invarian architecture for learning sets, ignoring all structural information (Zaheer jet al, 2017). GraphMLP basically extends DeepSet by incorporating graph structure during training neighborhood contrastive loss, which encourages linked nodes to have similar representa- tions (Hu et al. 202T).
Graph Convolutional Network (GCN) The GCN is a foundational GNN architecture that learss node representations by efficiently aggregating feature information from its immediate neighbors through a spectral-based graph convolution (Kipf & Welling 2017).
GraphSAGE Short for Graph SAmple and aggreGatE, this model provides a framework for inductive node embedding (Hamilton et al. 2017). Instead of training the entire graph, it learns
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aggregation functions on a fixed-size sample of node's neighborhood, allowing it to generalize to unseen nodes and graphs.
Graph Isomorphism Network (GIN) The GIN is powerful GNN designed be discriminative as the Weisfeiler-Lehaal (WL) graph isomorphism test (Xu et al. 2019). It achieves this by using an MLP to update node features, making it highly effective for tasks requiring a strong understanding of graph structure.
Graph Attention Network v2 (GATv2) GATv2 (Brody et al. 2022) an improved version of the original GAT model (Veličković et al. 2018) that uses modified attention mechanism to make it strictly more expressive. By assigning different importance weights to different nodes in a neighborhood, both GAT and GATv2 can focus on the most relevant parts of the graph for given
task,
Neural Sheaf Diffusion (NSD) NSD generalizes message passing cellular sheaves, which are topological structures capable of representing more complex relationships (Bodnar et al. 2022). This allows capture richer structural and relational information than standard GNNs.
TopoTune TopoTune is a framework designed to systematically generalize any GNN into topo- logical neural network, making higher-order structures accessible for learning (Papillon et al. 2025). It operates by taking GNN input and using it building block within architecture called a Generalized Combinatorial Complex Network (GCCN). This is achieved by expanding higher-order structure (like simplicial or cell complex) into a collection of graphs, which are then processed by an ensemble of synchronized GNNs. This approach democratizes topological deep learning by allowing practitioners casily "upgrade" existing GNNs reason about complex, multi-way relationships beyond simple edges.
expressive
GPS (Graph Transformer) The GPS model combines the expressive power transformers with standard message-passee GNNs (Rampášek et al., 2022).. By integrating local structural information with global attention mechanisms and positional encodings, it aims to capture a wide range of dependencies in the graph, making it very powerful and flexible architecture.
M.3 HYPERPARAMETES OPTIMIZATION
For each model, we leverage the TopoBench infrastructure (ellytnikkvv et al. 2025) to perform an extensive grid search identify optimal hyperparameter configurations, optimizing for the highest mean accuracy on held-out validation set (over three dataset seeds). Table 7 details the complete search space used for every hyperparameter of each model (following Topo logic of feature encoder, backbone and readout modules), providing basis for the reproducibility of our experiments. Unless otherwise specified in Table 7 we note that sum is the by default pooling method.
Remark. It should be noted that these spaces were refined based on preliminary, larger-scale grid searches; we pruned parameter options that consistently showed negligible or detrimental impact on performance focus the most influential hyperparameters. Furthermore, to limit the combinatorial explosion of the search space, parameters such as batch size and optimizer settings were fixed. These values were informed by previous benchmarks in TopoBench and TopoTTune (Papillon al., 2025): for example, we used the Adam optimizer with a learning rate of 0.001 and set the batch size to 32 for all inductive experiments.
M.4 HARDWARE DETAILS
The hyperparameter search is executed on Linux machine with 256 cores, 1TB of system memory, and 4 NVIDIA H100 GPUs, each with 94GB of GPU memory.
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Table 6: Default GraphUniverse generation parameters for Inductive and Transductive settings. Differences are highlighted in bold.
Inductive Value
Transductive Value
Universe Parameters
Number of communities (K)
Feature Dimension
15
15
Center Variance (a (oggter)
Cluster Variance (a (iluser
0.5
0.5
Edge Propensity Variance (c)
Seed
42
42
Family Parameters
Number of graphs
1000
Min Node Count (Nmin)
50
1000
Max Node Count (Ammax)
200
1000
Min Communities (kmin)
4
10
Max Communities (kmax)
Homophily Range (hmins kmax)
[0.4, 0.6]
[0.5, 0.5]
Average Degree Range (dmin: dmax))
[1.0, 5.0]
[2.5, 2.5]
Degree Separation Range (pmin: Prmxx)
[0.5, 0.81
[0.5, 0.51
Degree distribution
power law
power law
Power Law Exponent Range (Cimins mmax)
[2.0, 2.5] (Inherited from Universe)
[2.5, 2.5]
Seed
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Table 7: Hyperparameter grid search space for each model.
Model
Hyperparameter
Search Space (Grid)
GCN
feature.encoder: wwttooddrruutttooooooooooooo ccaannels
{32, 64)
backbone.num namers
{2, 4)
backbeme.dropout
(0.2, 0.4]
{[16), []}
GIN
feature.encoder.or
channeels
{32, 64)
{2. 4)
backbone.dropout
(0.2, 0.4)
readout.hicden_layers
{116), []]
GraphSAGE
(esturee.encoerr..ott.ccaaaee
(32, 64]
backbone.nnm
(2,4)
backbone.dzopout
(0.2, 0.4)
readout.hidddm.layerr
{[16]. [}}
GAT
fecture.enccerr..tt.ccaaaaa.
(32, 64)
backbore.num,laayer.
(2
backbome..heas
(2, 4,4,8
backbene
(..0.
readout.hioden.lyerrs
{[16), []
GPS
meatureeencodez..ttt.ccammee
(32, 64)
backbone.num,layyen
(2,4)
backbone.heads
(4)
backbone.dropout
{0.2, 0.4}
backbone.atth.tyyp
(multihead, performer) (RWSE, LapPE)
transforms.ennodding
readout..i.dden.aayerr
[[16]. [I]
NSD
feature.encoder..ott.chmmm
(32, 64)
backbone.
4, 6]
backbone.dropott
(.22, 0.4]
backbone.sheaf..ty.
[bundle, diag)
transforms.
(RWSE, LapPE}
readeatthhiceenlayere
{[16), [I]
GraphMLP
featureeniccerr..tttchameee
(32, 64)
backbeme.order.
{2.
backbone.dropon
(0.2, 0.4)
readout..hidden.ayyee
{[16]. []}
DeepSet
feature.anncdee..ot..ccammell
(32, 64]
readout.hiodde_layerr
[164, 32), [32, 16), [16]] 102,0..4
readout.dropout
Tono
model type
I
feature.encoder out.channels
32, 64)
tune-gan backbone. layers
(GCN, GIN, GAT, GraphSAGE] [2,4)
readout. ing type backbone wookbone.neieee neighbrrooss
mean, sum)
[10 predefined topological operator sets)
32


Reviews:
[{"review_id": "3XrbxhnGid", "reviewer": "Reviewer_Wz29", "summary": "This paper introduces GraphUniverse, a framework for generating families of synthetic graphs to systematically study inductive generalization in graph learning. By extending the Degree-Corrected Stochastic Block Model into a hierarchical generator, GraphUniverse produces semantically consistent graphs with controllable properties such as homophily, degree, and feature variance. Experiments show that model rankings differ substantially between transductive and inductive settings, revealing key insights into robustness and generalization behavior across diverse graph architectures.", "questions": "* Can the authors include a heterophily-specialized model (e.g., FAGCN, GPR-GNN, or H2GCN) in their comparisons, specifically for experiments varying homophily? This addition could better highlight how such models are specialized toward particular regimes by showing opposite performance trends across low- and high-homophily settings.  \n* Could the authors include a discussion of recent graph foundation model efforts that have demonstrated the benefits of using synthetic data, such as GraphFM \\[1\\], KumoRFM \\[2\\], and OpenGraph \\[3\\]? Including a brief overview of these works, along with parallels to how synthetic data has driven major advances in other domains, would strengthen the paper’s positioning and broaden its appeal to the wider graph learning and foundation model community.  \n* Can the authors add an analysis similar to Figures 1 and 2 from GraphWorld \\[4\\] to illustrate where real-world graph datasets lie within the GraphUniverse parameter space?\n\n\\[1\\] Lachi, Divyansha, et al. \"Graphfm: A scalable framework for multi-graph pretraining.\" *arXiv preprint arXiv:2407.11907* (2024).  \n\\[2\\] Fey, M., Kocijan, V., Lopez, F., Lenssen, J. E., & Leskovec, J. KumoRFM: A Foundation Model for In-Context Learning on Relational Data.  \n\\[4\\] Xia, L., Kao, B., & Huang, C. (2024). Opengraph: Towards open graph foundation models. *arXiv preprint arXiv:2403.01121*.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "* Fixes a fundamental limitation of GraphWorld and other synthetic graph generators by enabling multi-graph, inductive evaluation, moving beyond the conventional single-graph, transductive paradigm.   \n* Demonstrates that transductive performance is not predictive of inductive generalization and reveals architecture-specific robustness patterns under controlled distribution shifts. The analyses provide clear empirical evidence for a well-known but rarely quantified observation in the community—that graph models are often dataset- and domain-specific, performing well only in certain structural regimes. GraphUniverse’s controlled setup brings this behavior to light in a direct and systematic way.  \n* The inclusion of an interactive demo application is a valuable and user-friendly addition, enhancing reproducibility and making the framework easy to explore and adopt for future research.", "weaknesses": "* While the proposed framework convincingly demonstrates inductive generalization on synthetic graph families, it remains unclear whether these findings extend to real-world datasets. The paper does not map real datasets into the GraphUniverse parameter space or provide empirical validation that inductive robustness measured on synthetic graphs predicts robustness in real settings. As a result, it is difficult to determine whether the generated families capture realistic structural regimes or how the observed trends translate to practical graph learning scenarios.  \n* The claim that GraphUniverse can facilitate or benefit graph foundation model pretraining feels premature given the absence of supporting experiments or evidence. While this is not the paper’s central claim, the authors should at least include a more substantive discussion clarifying how GraphUniverse-generated data could be integrated into large-scale pretraining pipelines or what aspects of the framework make it suitable for such use.  \n* The underlying DC-SBM formulation assumes discrete, non-overlapping communities, whereas many real-world graphs contain overlapping structures and higher-order motifs that this model cannot capture. \n\nMy main concern is whether the insights obtained from GraphUniverse will transfer to real-world data. I suggest that the authors evaluate whether model rankings observed in the synthetic setting align with those on real benchmarks. For example, testing models on both heterophilic and homophilic real-world datasets could reveal whether performance trends from GraphUniverse are predictive of behavior in practical scenarios. It would be even more compelling if the authors also computed model rankings using GraphWorld and demonstrated that the rankings from GraphUniverse align more closely with those from real data. This would provide strong evidence for the external validity and practical relevance of the proposed framework. If the authors can demonstrate this, I am willing to increase my score.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZJ11vfpsVm", "reviewer": "Reviewer_XXiP", "summary": "This paper addresses the challenge of evaluating the true generalization capability of graph learning models by proposing the GraphUniverse framework. This framework systematically assesses model performance on unseen graph data by generating controllable graph families with semantically consistent communities. Experiments demonstrate that strong performance on known graphs does not guarantee generalization ability, and a model's robustness to distribution shifts heavily depends on the structural characteristics of the training graphs (such as the level of homophily in node connections), thus challenging the conventional practice of judging model quality solely based on transductive performance", "questions": "1. How can the authors ensure that the paper's central finding—the gap between transductive and inductive performance—is not an artifact of the DC-SBM generator itself? Could models not designed for community-structured data be unfairly penalized by this evaluation framework?\n\n2. Link prediction is a key inductive graph learning task. Why was it omitted from the benchmark? Can the GraphUniverse framework be extended or adapted to support a systematic evaluation of inductive link prediction?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "1. The work directly confronts the \"flawed benchmarking culture\" in graph learning—an over-reliance on a few static benchmarks and transductive evaluation, which is considered a major bottleneck hindering progress in the field.\n\n2. GraphUniverse is the first framework to enable large-scale, systematic evaluation of inductive generalization. Its paradigm of generating graph families with persistent semantics, rather than single graphs, is a significant advancement over existing tools like GraphWorld.\n\n3. The experiments compellingly demonstrate the gap between transductive and inductive performance and reveal that model robustness is context-dependent. not universal. These findings have important implications for model evaluation and development in the field.\n\n4. The paper is accompanied by an interactive demo and a promise to open-source the package. This is a valuable public contribution that will greatly facilitate future research.", "weaknesses": "1. The entire framework is built upon the Degree-Corrected Stochastic Block Model (DC-SBM), which is inherently biased toward generating graphs with strong community structures. This limits the applicability of the framework's conclusions to other common graph topologies that lack clear communities (e.g., geometric graphs, some biological networks), thus constraining its universality.\n\n2. The experimental evaluation focuses primarily on node-level community detection and a simple graph-level task (triangle counting). It notably omits other critical graph learning tasks, especially Link Prediction, which is itself a core research area for inductive learning. This narrow task scope weakens the generality of the paper's main conclusions.\n\n3. In light of the two points above, the paper's claim of enabling \"systematic evaluation of inductive generalization\" may be too broad. The findings and conclusions are largely about generalization for \"node classification tasks\" on \"graphs with community structure,\" a scope limitation that should be more explicitly stated in the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "RQbZyhy0Nz", "reviewer": "Reviewer_N9mv", "summary": "This paper presents a framework to generate graphs to evaluate graph learning models on transductive and inductive tasks. Furthermore, the authors use their tool to benchmark a variety of GNNs and achieve results about the impact of data on model generalization (e.g. the impact of distribution shifts).", "questions": "- See weaknesses\n\n\n**Overall,** I think this is a good paper with clear flaws in its presentation. Thus, I vote for (6 - marginally above acceptance threshold).\n\n### Miscellaneous\n- Section 4.1.1 should in my opinion be a subsection instead of the only sub-subsection in the main paper.\n- No information about the pooling type for some architectures.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- **(S1) Originality + Significance:** Current benchmarks of GNNs are severely lacking (as pointed out by _[Position: Graph learning will lose relevance due to poor benchmarks](https://arxiv.org/abs/2502.14546)_). GraphUniverse comes timely to help with this issue. It seems to be easy to use (please contrast with W3) and can be used to create a diverse array of graphs. This could lead the community to a more principled evaluation of GNN architectures and thus to a better understanding of their capabilities.\n- **(S2) Clarity:** The paper is well written and easy to understand. (Please contrast with W2)\n- **(S3) Quality:** The authors use GraphUniverse to benchmark a variety of different GNNs. I am impressed by the variety of benchmarked GNNs and find the results interesting:\n\t- GIN struggles in the transductive setting (but not the inductive) whereas the graph transformer performs similarly in both settings. This is interesting to me, as I would have expected GIN to also perform well in the inductive case. Furthermore, while the explanation _... indicating its success may stem from memorizing a single graph’s structure_ makes intuitive sense, it goes against my experience of GIN performing well on inductive molecular tasks.\n\t- Furthermore, the benchmarks show that models are less robust to distribution shifts than thought. This opens the research direction of developing models that perform better under distribution shifts.", "weaknesses": "I like the contributions of this work and my weaknesses all center around the presentation.\n\n- **(W1) Citations:** This paper often cites the arXiv version of papers instead of the published version. While not only formally wrong, this makes it difficult to evaluate whether this paper builds upon established research or on irrelevant work.\n- **(W2) Presentation of experimental results:** Figure 3 is the cornerstone of the experimental section. It contains almost all results and most research questions reference it. Despite this, Figure 3 is extremely unclear making it difficult to verify the claims of the authors. Below are some pointers for improvement that could serve as inspiration to make the figure more legible:\n\t- The first line reads \"Transductive\" and the second \"Inductive\". However, in the legend you have a separate style for inductive and transductive results and according to this the first result in the transductive line is inductive.\n\t- The split into (A) and (B) is not that clear. It would help if you could explicitly mention in the figure (not in the caption!) which research question (A) and (B) answer.\n\t- Explicit color coding and re-use: When your main text refers to the result of a model in Figure 3 re-use the color of the model in the main text. This makes it easier to find that information in the figure. Furthermore, you could explicitly color code types of models (which I think you attempted but it is not clear enough).\n\t- Why do some results have N/A?\n\t- Alternatively, you could simplify this figure by showing the results of fewer models (and putting the more extensive figure in the appendix). As far as I can see, you are not discussing the results of GCN, GAT and TopoTune in the main text anyway.\n- **(W3) No code:** While the authors pledge to make the code public there is no code for the reviewers to check. It is unclear how easy to use GraphUniverse is. Furthermore, I wanted to check the code for the following. You write:\n>Graph-level tasks (triangle counting) initially show only GPS, NSD, and GIN effectively solving the task, but while GPS and NSD maintain performance when scaling to larger graphs, GIN fails to generalize, suggesting traditional MPNNs overfit to training graph sizes.\n\n\n- **(W4) Figures of graphs:** Fundamentally, this paper is about generating graph data. Thus, the  paper should include some figures of generated graphs. While the interactive demo is great, it is not guaranteed to be online or work in the future.\n\nIs it possible that these models differ in how they aggregate local embeddings to the global level (e.g. sum vs mean pooling)?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ha3ea8Jk7j", "reviewer": "Reviewer_3ZL3", "summary": "GraphUniverse is a hierarchical generator for families of graphs designed to study inductive generalization across unseen graphs. At the “universe” level it fixes persistent communities with heterogeneous edge propensities, degree profiles, and feature centroids; “family” ranges then constrain homophily, average degree, size, and degree separation; graph instances sample from these ranges and realize edges via a Bernoulli reformulation of DC‑SBM with community‑coupled degree factors and feature Gaussians. Validation correlates generation parameters with graph properties, signal strength, and cross‑graph consistency. Benchmarking compares inductive vs. transductive performance and robustness under controlled distribution shifts and size scaling for several architectures.", "questions": "Please see weakness", "limitations": "", "rating": 4, "confidence": 5, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. A very timely and important direction for the whole graph community\n2. the demo website is well designed and validates the reproducibility\n3. comprehensive experiments", "weaknesses": "1. the biggest problem of this framework is that I don't see its clear advantage over other generative frameworks like GraphWorld, CGT (Graph Generative Model for Benchmarking Graph Neural Networks, icml 2023). And it doesn't mention a very relevant work, A Metadata-Driven Approach to Understand Graph Neural Networks (nips 2023). Although authors conduct comprehensive experiments on using this framework to evaluate different kinds of GML architectures, two experiments are missing: (1) the horizontal comparison between this framework against other generative pipelines, this makes the value of this work unclear; (2) This paper presents train on syn, test on real-world experiments, but there's no train on syn, test on real-world ones. This limits the usage of these synthetic datasets. I also doubt whether these purely synthetic graphs can really be used to pressure test GNN models. I personally think the way CGT is adopting (start from real-world ones and do generation) is more promising. \n2. In the very beginning, authors talk about the problem of graph foundation model. However, i don't see this work able to address these relevant problems. \n3. Statements about enabling “next‑generation graph foundation models” are aspirational without pre‑training results (Abstract; Sec. 1, 6). No direct evidence found in the manuscript.\n4. Semantic “consistency” is validated via statistical proxies, not via downstream cross‑family transfer beyond the reported tasks (Fig. 2; Sec. 4.2–5).\n5. Why are these factors considered? There's no validation. You can see (Metadata-Driven Approach to Understand Graph Neural Networks (nips 2023)) conducts a statistical test, which is much more promising and scientific. \n6. Table 1 reports only two sizes (100 vs. 1000 nodes) and single‑thread CPU throughput; linearity claims rest on a very small sample (Sec. 4.3; Table 1)—limited generality.\n7. Minor notation/typo inconsistencies (e.g., “DCCC‑SBM true true” in Table 4)\n8. Only one transformer variant (GPS) is included (Sec. 5.1), narrowing architectural coverage. GraphGPS also heavily relies on GNN.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)