RESEARCH PROJECT · HYPERGRAPH GENERATION

Hypergraph Generation
with Latent Diffusion

Janus: Hypergraph Generation with Latent Diffusion, illustrated with a two-faced Janus and hypergraph structures

Modeling nodes and hyperedges as two coupled generative views.

Valerio Di Pasquale 1 Alessia Antelmi2Mirko Polato2 Carmine Spagnuolo 1
1 University of Salerno2 University of Turin
↗ Paper (NeurIPS 2026)⌘ Code

Generate higher-order structure, not just pairwise edges.

Real-world interactions often involve groups of entities at once. Janus learns a distribution of hypergraphs by combining a dual-view variational autoencoder with cross-conditioned latent diffusion, preserving the relationship between nodes and hyperedges.

Janus framework: sub-hypergraph sampling, dual-view encoding, coupled diffusion, decoding and assembly.
Janus framework: sub-hypergraph sampling, dual-view encoding, coupled diffusion, decoding and assembly.
01

Dual-view representation

Learn separate but interdependent node and hyperedge latent spaces with a β-VAE.

02

Coupled diffusion

Generate two latent views with cross-attention, then decode a coherent incidence structure.

03

Local-to-global generation

Sample local sub-hypergraphs and merge them into realistic larger structures.

How Janus works

1

Decompose the observed hypergraph

Extract overlapping sub-hypergraphs to learn from a single high-order network without modeling the combinatorial hyperedge space directly.

2

Learn two aligned latent spaces

The β-VAE encodes node and hyperedge representations and learns to recover their incidence relations.

3

Diffuse with cross-view conditioning

Paired diffusion transformers denoise both latent streams while exchanging information through cross-attention.

4

Decode and assemble

Decode synthetic sub-hypergraphs and reconcile entities for full hypergraph generation. Janus generates nodes and hyperedges; JanusNC works with a prescribed node set.

Cross-attention denoising block.
Cross-attention denoising block.
Dual-view variational autoencoder.
Dual-view variational autoencoder.

A novel Hypergraph Generation Evalutation Framework

Micro

Node degree, hyperedge size and clustering-coefficient distributions.

Meso

Community structure, degree assortativity and open/closed triangles.

Macro

Global size, connectivity, transitivity, overlapness and modularity.

Higher-order

Hyper NetSimile, Hyperedge Portrait Divergence and reconstruction under fixed nodes.

Taxonomy of the evaluation framework.
Taxonomy of the evaluation framework.

Faithful across structural scales

Evaluated on five real-world datasets against nine baseline methods, using microscopic, mesoscopic, macroscopic and higher-order similarity metrics.

5 real datasets9 baselines2 generation settings5 samples / metric
Average-rank summary of generation quality across datasets and metric families.
Average-rank summary of generation quality across datasets and metric families.

Janus performs consistently across scales, with particularly strong mesoscopic organization and higher-order structural similarity. Some local metrics on email-Eu remain challenging; full per-dataset tables are available below.

0.19
Hyper NetSimile · CHS

JanusNC versus 2.07 for HyperPLR. Lower is better.

0.0038
CECC error · CPS

JanusNC versus 0.0292 for HyperLap. Lower is better.

0.9215
Hyperedge Jaccard · Enron

JanusNC versus 0.1613 for HyperPLR. Higher is better.

Why cross-attention matters

Removing cross-attention increases Hyper Portrait Divergence from 0.1158 to 0.1924, and Hyper NetSimile from 0.8984 to 1.6320 on the email-Enron ablation. Coupled denoising helps maintain high-order structure.

Supplementary Results

Detailed numerical results from the paper and supplementary material. Expand a category, then a table to inspect the results.

Model hyperparameters2 tables
View result table
Size # Layers # Heads Hidden size Latent dim.
S 1 2 128 128
M 2 4 256 128
View result table
Size # Layers # Heads Hidden size Latent dim.
S 4 2 256 128
M 8 4 512 128
Dataset statistics1 table
View result table
Abbreviation Dataset |V| |E| # Connected Components
NDC NDC-classes 1,161 1,088 183
CHS contact-high-school 327 7,818 1
CPS contact-primary-school 242 12,704 1
EE email-Enron 143 1,512 1
EEu email-Eu 998 25,027 20
Detailed benchmark results43 tables
local statistics NDC-classes

Microscopic local statistics performance comparison on the NDC-classes dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Avg. Clustering (lower is better), Node Degree Dist. (lower is better), Hyperedge Size Dist. (lower is better).

Model Clique Exp. Avg. Clustering Node Degree Dist. Hyperedge Size Dist. Avg. Rank
5cunconstrained
SubsetSampling 0.0888 ± 0.0096 4.7782 ± 0.8253 0.1899 ± 0.1359 1.6667
Janus 0.0460 ± 0.0109 5.0146 ± 0.6866 1.5250 ± 0.2165 2.6667
HyperPA 0.1331 ± 0.0131 6.8262 ± 0.7726 0.1471 ± 0.0414 3.0000
TheRA 0.1817 5.7460 0.3604 3.3333
NaivePA 0.2025 ± 0.0118 6.5787 ± 0.8933 0.9076 ± 0.1805 4.3333
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0003 ± 0.0004 0.0215 ± 0.0149 0.0099 ± 0.0050 1.0000
Janus--VAE 0.0018 ± 0.0010 0.1602 ± 0.0184 0.0398 ± 0.0183 2.0000
5cnode-set constrained
HyperDK11 0.1414 ± 0.0040 0.1874 ± 0.0119 0.1587 ± 0.0049 2.0000
JanusNC 0.0485 ± 0.0046 1.3110 ± 0.0678 0.5228 ± 0.0487 2.3333
Hyperlap 0.0607 ± 0.0104 1.9195 ± 0.1087 0.2440 ± 0.0000 2.8333
Hyperlap+ 0.1388 ± 0.0052 2.2368 ± 0.0595 0.2440 ± 0.0000 3.5000
HyperPLR 0.1796 ± 0.0098 1.3261 ± 0.1043 1.1018 ± 0.0358 4.3333
HyperDK00 0.3555 ± 0.0034 5.2728 ± 0.0411 2.0094 ± 0.0227 6.0000
local statistics contact-high-school

Microscopic local statistics performance comparison on the contact-high-school dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Avg. Clustering (lower is better), Node Degree Dist. (lower is better), Hyperedge Size Dist. (lower is

Model Clique Exp. Avg. Clustering Node Degree Dist. Hyperedge Size Dist. Avg. Rank
5cunconstrained
Janus 0.0315 ± 0.0048 2.9524 ± 0.4882 0.0690 ± 0.0023 2.0000
SubsetSampling 0.0128 ± 0.0120 27.4937 ± 1.9276 0.0168 ± 0.0044 2.0000
HyperPA 0.1057 ± 0.0056 31.4689 ± 1.5729 0.0037 ± 0.0019 2.6667
TheRA 0.1493 13.5076 0.2823 3.6667
NaivePA 0.1352 ± 0.0127 40.6730 ± 2.4702 0.9990 ± 0.0084 4.6667
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0003 ± 0.0004 0.1058 ± 0.0195 0.0016 ± 0.0004 1.0000
Janus--VAE 0.0020 ± 0.0004 0.4606 ± 0.0672 0.0070 ± 0.0009 2.0000
5cnode-set constrained
JanusNC 0.0074 ± 0.0023 0.7314 ± 0.0568 0.0245 ± 0.0007 2.0000
Hyperlap+ 0.1287 ± 0.0053 1.2159 ± 0.1165 0.0000 ± 0.0000 2.5000
Hyperlap 0.1625 ± 0.0039 1.0920 ± 0.0585 0.0000 ± 0.0000 2.8333
HyperPLR 0.0325 ± 0.0004 11.1189 ± 0.2333 0.4360 ± 0.0030 4.0000
HyperDK11 0.2099 ± 0.0020 2.5694 ± 0.0745 0.0156 ± 0.0007 4.3333
HyperDK00 0.1614 ± 0.0032 16.4820 ± 0.1588 0.7773 ± 0.0165 5.3333
LaTeX table source ↗
local statistics contact-primary-school

Microscopic local statistics performance comparison on the contact-primary-school dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Avg. Clustering (lower is better), Node Degree Dist. (lower is better), Hyperedge Size Dist. (lower

Model Clique Exp. Avg. Clustering Node Degree Dist. Hyperedge Size Dist. Avg. Rank
5cunconstrained
Janus 0.0084 ± 0.0020 5.5625 ± 0.1461 0.0311 ± 0.0032 1.6667
HyperPA 0.0261 ± 0.0131 67.0734 ± 0.6297 0.0059 ± 0.0056 2.0000
TheRA 0.0799 22.9917 0.3481 3.0000
SubsetSampling 0.1564 ± 0.0189 70.6643 ± 2.9423 0.0067 ± 0.0013 3.3333
NaivePA 0.2096 ± 0.0097 86.0562 ± 5.0824 0.9974 ± 0.0074 5.0000
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 0.1810 ± 0.0486 0.0012 ± 0.0002 1.0000
Janus--VAE 0.0009 ± 0.0004 1.0091 ± 0.0713 0.0063 ± 0.0004 2.0000
5cnode-set constrained
JanusNC 0.0038 ± 0.0015 2.1124 ± 0.0861 0.0219 ± 0.0009 1.6667
Hyperlap 0.0292 ± 0.0009 2.2926 ± 0.3851 0.0000 ± 0.0000 1.8333
Hyperlap+ 0.1290 ± 0.0229 4.3504 ± 1.1593 0.0000 ± 0.0000 3.1667
HyperDK11 0.0686 ± 0.0019 10.7215 ± 0.2688 0.0415 ± 0.0012 4.0000
HyperPLR 0.0345 ± 0.0000 16.0023 ± 0.4694 0.2867 ± 0.0041 4.3333
HyperDK00 0.1923 ± 0.0027 37.8463 ± 0.4755 0.7498 ± 0.0061 6.0000
LaTeX table source ↗
local statistics email-Enron

Microscopic local statistics performance comparison on the email-Enron dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Avg. Clustering (lower is better), Node Degree Dist. (lower is better), Hyperedge Size Dist. (lower is better).

Model Clique Exp. Avg. Clustering Node Degree Dist. Hyperedge Size Dist. Avg. Rank
5cunconstrained
Janus 0.0112 ± 0.0054 3.5287 ± 0.3674 0.3067 ± 0.0390 1.6667
HyperPA 0.0734 ± 0.0403 11.7011 ± 1.3787 0.0616 ± 0.0376 2.0000
TheRA 0.0805 10.6294 0.3776 3.0000
SubsetSampling 0.1391 ± 0.0114 13.8181 ± 1.5490 0.0883 ± 0.0253 3.6667
NaivePA 0.1589 ± 0.0112 12.8113 ± 2.5531 0.9956 ± 0.0580 4.6667
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0004 ± 0.0009 0.0448 ± 0.0607 0.0024 ± 0.0033 1.0000
Janus--VAE 0.0014 ± 0.0016 0.0937 ± 0.0663 0.0035 ± 0.0027 2.0000
5cnode-set constrained
JanusNC 0.0021 ± 0.0018 0.8478 ± 0.1101 0.0493 ± 0.0064 1.3333
HyperDK11 0.0414 ± 0.0052 1.5804 ± 0.1058 0.0513 ± 0.0095 2.3333
Hyperlap 0.0524 ± 0.0087 2.5836 ± 0.3062 0.1217 ± 0.0000 3.5000
HyperPLR 0.0010 ± 0.0009 3.2313 ± 0.1835 0.4610 ± 0.0112 3.6667
Hyperlap+ 0.2479 ± 0.0132 2.6126 ± 0.5885 0.1217 ± 0.0000 4.5000
HyperDK00 0.0868 ± 0.0029 14.4014 ± 0.2962 0.6538 ± 0.0255 5.6667
local statistics email-Eu

Microscopic local statistics performance comparison on the email-Eu dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Avg. Clustering (lower is better), Node Degree Dist. (lower is better), Hyperedge Size Dist. (lower is better). Co

Model Clique Exp. Avg. Clustering Node Degree Dist. Hyperedge Size Dist. Avg. Rank
5cunconstrained
SubsetSampling 0.1597 ± 0.0130 30.5596 ± 5.3939 0.0275 ± 0.0073 1.6667
NaivePA 0.2368 ± 0.0048 49.8866 ± 2.2798 0.9942 ± 0.0263 2.6667
HyperPA 0.3406 ± 0.0116 81.1249 ± 0.6443 0.0181 ± 0.0052 3.3333
Janus 0.1319 ± 0.0023 64.0602 ± 0.7441 2.1892 ± 0.0087 3.3333
TheRA 0.3454 62.4391 1.2707 4.0000
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0004 ± 0.0003 0.6631 ± 0.1053 0.0135 ± 0.0027 1.0000
Janus--VAE 0.0030 ± 0.0025 3.8457 ± 0.2356 0.0843 ± 0.0067 2.0000
5cnode-set constrained
HyperDK11 0.1710 ± 0.0047 2.9966 ± 0.1062 0.0803 ± 0.0025 2.0000
Hyperlap 0.1310 ± 0.0023 9.4033 ± 0.5780 0.1950 ± 0.0000 2.8333
HyperPLR 0.0455 ± 0.0025 6.4337 ± 0.5677 0.9769 ± 0.0014 3.0000
JanusNC 0.0286 ± 0.0059 23.5887 ± 0.5009 0.7265 ± 0.0072 3.3333
Hyperlap+ 0.2936 ± 0.0044 9.9931 ± 0.4616 0.1950 ± 0.0000 4.1667
HyperDK00 0.2261 ± 0.0010 71.7068 ± 0.1027 0.9839 ± 0.0109 5.6667
mesoscopic · community NDC-classes

Mesoscopic community performance comparison on the NDC-classes dataset. Methods are grouped by kind. Reported metrics: Normalized Mutual Information (higher is better). Colored cells mark the top three values within each kind for ranked metrics (gree

Model Normalized Mutual Information Avg. Rank
3c(Janus) reconstruction upper bound
JanusNC--VAE 0.8035 ± 0.0597 1.0000
Janus--VAE 0.5945 ± 0.0903 2.0000
3cnode-set constrained
JanusNC 0.4520 ± 0.0104 1.0000
HyperPLR 0.3794 ± 0.0107 2.0000
HyperDK00 0.2567 ± 0.0017 3.0000
HyperDK11 0.2466 ± 0.0094 4.0000
Hyperlap+ 0.2382 ± 0.0014 5.0000
Hyperlap 0.1882 ± 0.0055 6.0000
LaTeX table source ↗
mesoscopic · community contact-high-school

Mesoscopic community performance comparison on the contact-high-school dataset. Methods are grouped by kind. Reported metrics: Normalized Mutual Information (higher is better). Colored cells mark the top three values within each kind for ranked metri

Model Normalized Mutual Information Avg. Rank
3c(Janus) reconstruction upper bound
JanusNC--VAE 0.8561 ± 0.0000 1.0000
Janus--VAE 0.6284 ± 0.3043 2.0000
3cnode-set constrained
JanusNC 0.1266 ± 0.0352 1.0000
HyperDK00 0.0414 ± 0.0053 2.0000
HyperDK11 0.0323 ± 0.0110 3.0000
HyperPLR 0.0289 ± 0.0055 4.0000
Hyperlap 0.0276 ± 0.0080 5.0000
Hyperlap+ 0.0219 ± 0.0039 6.0000
LaTeX table source ↗
mesoscopic · community contact-primary-school

Mesoscopic community performance comparison on the contact-primary-school dataset. Methods are grouped by kind. Reported metrics: Normalized Mutual Information (higher is better). Colored cells mark the top three values within each kind for ranked me

Model Normalized Mutual Information Avg. Rank
3c(Janus) reconstruction upper bound
JanusNC--VAE 0.6708 ± 0.1868 1.0000
Janus--VAE 0.3403 ± 0.3302 2.0000
3cnode-set constrained
JanusNC 0.1207 ± 0.0400 1.0000
HyperDK00 0.0608 ± 0.0231 2.0000
HyperDK11 0.0404 ± 0.0078 3.0000
HyperPLR 0.0365 ± 0.0048 4.0000
Hyperlap+ 0.0248 ± 0.0095 5.0000
Hyperlap 0.0233 ± 0.0062 6.0000
LaTeX table source ↗
mesoscopic · community email-Enron

Mesoscopic community performance comparison on the email-Enron dataset. Methods are grouped by kind. Reported metrics: Normalized Mutual Information (higher is better). Colored cells mark the top three values within each kind for ranked metrics (gree

Model Normalized Mutual Information Avg. Rank
3c(Janus) reconstruction upper bound
JanusNC--VAE 0.7338 ± 0.0527 1.0000
Janus--VAE 0.7131 ± 0.1132 2.0000
3cnode-set constrained
JanusNC 0.1417 ± 0.0384 1.0000
HyperDK11 0.0839 ± 0.0241 2.0000
HyperPLR 0.0821 ± 0.0160 3.0000
HyperDK00 0.0723 ± 0.0209 4.0000
Hyperlap+ 0.0599 ± 0.0206 5.0000
Hyperlap 0.0564 ± 0.0167 6.0000
LaTeX table source ↗
mesoscopic · community email-Eu

Mesoscopic community performance comparison on the email-Eu dataset. Methods are grouped by kind. Reported metrics: Normalized Mutual Information (higher is better). Colored cells mark the top three values within each kind for ranked metrics (green b

Model Normalized Mutual Information Avg. Rank
3c(Janus) reconstruction upper bound
JanusNC--VAE 0.4873 ± 0.1893 1.0000
Janus--VAE 0.0754 ± 0.0317 2.0000
3cnode-set constrained
JanusNC 0.0562 ± 0.0064 1.0000
HyperDK11 0.0378 ± 0.0033 2.0000
HyperPLR 0.0371 ± 0.0012 3.0000
HyperDK00 0.0320 ± 0.0017 4.0000
Hyperlap+ 0.0291 ± 0.0031 5.0000
Hyperlap 0.0272 ± 0.0022 6.0000
mesoscopic · structural patterns NDC-classes

Mesoscopic structural patterns performance comparison on the NDC-classes dataset. Methods are grouped by kind. Reported metrics: Hypergraph Degree Assortativity (lower is better), # Closed Triangles (lower is better), # Open Triangles (lower is bette

Model Hypergraph Degree Assortativity # Closed Triangles # Open Triangles Avg. Rank
5cunconstrained
Janus 0.1666 ± 0.0454 1638.8000 ± 1184.8674 1229.2000 ± 726.4384 1.3333
TheRA 0.1268 63676.0000 2717.0000 2.6667
HyperPA 0.5059 ± 0.0100 9521.2000 ± 2744.1732 4590.0000 ± 864.6465 3.3333
SubsetSampling 0.4480 ± 0.0130 42097.0000 ± 5611.2747 45282.2000 ± 7923.7922 3.3333
NaivePA 0.4508 ± 0.0101 58365.2000 ± 9078.8307 83767.2000 ± 16902.7303 4.3333
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0669 ± 0.0171 19.2000 ± 31.8700 6.4000 ± 9.7365 1.3333
Janus--VAE 0.0392 ± 0.0305 349.4000 ± 136.0121 60.2000 ± 76.0375 1.6667
5cnode-set constrained
JanusNC 0.0450 ± 0.0168 69.4000 ± 53.8359 11.6000 ± 14.2934 1.0000
HyperPLR 0.2684 ± 0.0040 4295.4000 ± 58.1016 7115.6000 ± 7.9875 2.3333
HyperDK00 0.4572 ± 0.0082 6289.2000 ± 471.9817 3508.8000 ± 280.3457 3.0000
Hyperlap 0.4122 ± 0.0150 81836.4000 ± 1011.5262 61507.8000 ± 3405.4202 4.6667
Hyperlap+ 0.4580 ± 0.0456 66595.0000 ± 1550.1950 19531.6000 ± 1953.4395 4.6667
HyperDK11 0.4794 ± 0.0034 60097.6000 ± 1117.6052 88940.6000 ± 1430.5832 5.3333
LaTeX table source ↗
mesoscopic · structural patterns contact-high-school

Mesoscopic structural patterns performance comparison on the contact-high-school dataset. Methods are grouped by kind. Reported metrics: Hypergraph Degree Assortativity (lower is better), # Closed Triangles (lower is better), # Open Triangles (lower

Model Hypergraph Degree Assortativity # Closed Triangles # Open Triangles Avg. Rank
5cunconstrained
Janus 0.0301 ± 0.0159 126.8000 ± 12.4780 4793.2000 ± 194.9172 1.3333
HyperPA 0.6444 ± 0.0363 269.4000 ± 115.0187 4506.4000 ± 3070.5346 2.6667
SubsetSampling 0.1427 ± 0.0241 4175.6000 ± 155.7524 19932.4000 ± 3405.5564 3.3333
TheRA 0.4547 2664.0000 18683.0000 3.3333
NaivePA 0.1781 ± 0.0154 11815.0000 ± 653.4635 190303.0000 ± 13624.2036 4.3333
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0199 ± 0.0237 9.4000 ± 3.7815 8.6000 ± 8.9051 1.3333
Janus--VAE 0.0176 ± 0.0121 72.8000 ± 13.3492 52.6000 ± 27.9517 1.6667
5cnode-set constrained
JanusNC 0.0186 ± 0.0109 7.4000 ± 4.2778 1744.2000 ± 107.8643 1.0000
HyperPLR 0.1022 ± 0.0271 3122.6000 ± 23.7023 13919.8000 ± 78.8365 3.0000
HyperDK11 0.1544 ± 0.0106 638.4000 ± 4.2190 47846.2000 ± 737.0853 3.3333
Hyperlap 0.1571 ± 0.0108 644.4000 ± 3.9115 19855.2000 ± 233.4603 4.0000
Hyperlap+ 0.2067 ± 0.0355 640.4000 ± 5.6391 31592.6000 ± 1138.7477 4.3333
HyperDK00 0.1561 ± 0.0047 13882.2000 ± 432.7519 164168.4000 ± 4677.3007 5.3333
LaTeX table source ↗
mesoscopic · structural patterns contact-primary-school

Mesoscopic structural patterns performance comparison on the contact-primary-school dataset. Methods are grouped by kind. Reported metrics: Hypergraph Degree Assortativity (lower is better), # Closed Triangles (lower is better), # Open Triangles (low

Model Hypergraph Degree Assortativity # Closed Triangles # Open Triangles Avg. Rank
5cunconstrained
Janus 0.0801 ± 0.0480 63.6000 ± 46.4898 7019.4000 ± 121.6955 1.0000
HyperPA 0.4062 ± 0.0626 392.4000 ± 162.4448 10614.8000 ± 6508.8942 3.0000
SubsetSampling 0.1410 ± 0.0288 3479.6000 ± 402.7975 17587.4000 ± 13833.6117 3.0000
TheRA 0.3084 3188.0000 23776.0000 3.6667
NaivePA 0.1654 ± 0.0128 17966.4000 ± 1516.3582 234592.8000 ± 35341.2282 4.3333
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0397 ± 0.0297 16.2000 ± 3.6332 34.6000 ± 36.6920 1.3333
Janus--VAE 0.0196 ± 0.0190 110.6000 ± 5.8566 472.6000 ± 139.8689 1.6667
5cnode-set constrained
JanusNC 0.0589 ± 0.0234 29.8000 ± 10.1833 3024.2000 ± 568.3350 1.3333
HyperPLR 0.0132 ± 0.0103 3094.8000 ± 52.2752 39809.8000 ± 52.2752 3.0000
Hyperlap+ 0.1667 ± 0.0083 159.0000 ± 49.4823 29547.6000 ± 15052.6589 3.3333
Hyperlap 0.1570 ± 0.0102 653.8000 ± 14.1315 41781.6000 ± 1997.6253 4.0000
HyperDK11 0.1560 ± 0.0060 835.2000 ± 6.3008 270809.0000 ± 2079.5510 4.3333
HyperDK00 0.1463 ± 0.0055 23663.8000 ± 649.0498 729303.0000 ± 9079.2175 5.0000
LaTeX table source ↗
mesoscopic · structural patterns email-Enron

Mesoscopic structural patterns performance comparison on the email-Enron dataset. Methods are grouped by kind. Reported metrics: Hypergraph Degree Assortativity (lower is better), # Closed Triangles (lower is better), # Open Triangles (lower is bette

Model Hypergraph Degree Assortativity # Closed Triangles # Open Triangles Avg. Rank
5cunconstrained
Janus 0.0225 ± 0.0173 115.2000 ± 111.5267 52.8000 ± 32.5223 1.0000
HyperPA 0.3035 ± 0.0161 2788.2000 ± 642.6027 3021.6000 ± 1355.3648 3.0000
SubsetSampling 0.1923 ± 0.0310 4195.6000 ± 947.8480 6291.6000 ± 1888.3376 3.0000
TheRA 0.0672 15588.0000 8665.0000 3.6667
NaivePA 0.2913 ± 0.0099 4252.0000 ± 998.4415 21891.2000 ± 3146.9657 4.3333
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0368 ± 0.0115 9.0000 ± 20.1246 4.8000 ± 10.7331 1.3333
Janus--VAE 0.0562 ± 0.0316 2.4000 ± 2.3022 6.8000 ± 8.1056 1.6667
5cnode-set constrained
JanusNC 0.0367 ± 0.0202 4.8000 ± 4.7645 81.8000 ± 24.8938 1.0000
HyperPLR 0.1523 ± 0.0124 1465.0000 ± 37.7624 2011.2000 ± 58.4953 2.6667
Hyperlap+ 0.2685 ± 0.0170 13682.8000 ± 758.4070 1323.6000 ± 308.0321 3.6667
HyperDK00 0.2760 ± 0.0156 208.4000 ± 114.8011 46651.0000 ± 838.4211 4.3333
Hyperlap 0.2677 ± 0.0115 16949.6000 ± 536.6850 15849.4000 ± 2582.9575 4.3333
HyperDK11 0.2845 ± 0.0091 4066.2000 ± 160.9323 37831.4000 ± 974.1536 5.0000
LaTeX table source ↗
mesoscopic · structural patterns email-Eu

Mesoscopic structural patterns performance comparison on the email-Eu dataset. Methods are grouped by kind. Reported metrics: Hypergraph Degree Assortativity (lower is better), # Closed Triangles (lower is better), # Open Triangles (lower is better).

Model Hypergraph Degree Assortativity # Closed Triangles # Open Triangles Avg. Rank
5cunconstrained
HyperPA 0.2943 ± 0.0130 68808.6000 ± 3983.7970 89072.8000 ± 13519.0941 2.3333
TheRA 0.0184 727153.0000 745023.0000 2.3333
SubsetSampling 0.1863 ± 0.0163 252536.0000 ± 14546.0439 1460210.8000 ± 200863.4878 2.6667
NaivePA 0.1972 ± 0.0088 314780.2000 ± 13796.5497 2009051.8000 ± 73805.9843 3.6667
Janus 0.0377 ± 0.0121 792608.6667 ± 27584.7766 2876772.6667 ± 41937.5387 4.0000
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0713 ± 0.0404 840.4000 ± 251.3708 1087.8000 ± 417.5598 1.3333
Janus--VAE 0.0649 ± 0.0464 17319.6000 ± 1658.2421 100531.6000 ± 8921.3966 1.6667
5cnode-set constrained
JanusNC 0.1381 ± 0.0422 2448.6000 ± 1077.1770 59049.4000 ± 1885.1407 1.0000
HyperPLR 0.2044 ± 0.0086 127367.4000 ± 155.7861 165975.6000 ± 1367.2133 3.0000
Hyperlap+ 0.1762 ± 0.0355 504741.4000 ± 11561.8625 673393.8000 ± 20347.2001 3.3333
HyperDK00 0.2067 ± 0.0023 5562.0000 ± 1384.6366 2467127.0000 ± 29901.9227 3.6667
Hyperlap 0.1974 ± 0.0065 718826.6000 ± 15624.4865 2837196.4000 ± 115211.6587 4.6667
HyperDK11 0.2148 ± 0.0007 417648.4000 ± 4489.0074 5279779.4000 ± 22907.1066 5.3333
LaTeX table source ↗
macroscopic · community NDC-classes

Macroscopic community performance comparison on the NDC-classes dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Modularity (lower is better), Hypergraph Modularity (lower is better), Bipartite Modularity (lower is better), Line Gr

Model Clique Exp. Modularity Hypergraph Modularity Bipartite Modularity Line Graph Modularity Avg. Rank
6cunconstrained
Janus 0.0715 ± 0.0122 0.1282 ± 0.0188 0.1147 ± 0.0175 0.1099 ± 0.0210 1.0000
TheRA 0.0938 0.1881 0.1516 0.1519 2.0000
SubsetSampling 0.4472 ± 0.0123 0.4062 ± 0.0297 0.3553 ± 0.0232 0.3567 ± 0.0236 3.0000
HyperPA 0.4777 ± 0.0098 0.5045 ± 0.0212 0.4701 ± 0.0126 0.4673 ± 0.0115 4.2500
NaivePA 0.5047 ± 0.0070 0.5370 ± 0.0173 0.4691 ± 0.0123 0.4685 ± 0.0139 4.7500
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0031 ± 0.0026 0.0059 ± 0.0028 0.0028 ± 0.0016 0.0016 ± 0.0011 1.2500
Janus--VAE 0.0064 ± 0.0062 0.0060 ± 0.0027 0.0017 ± 0.0009 0.0019 ± 0.0009 1.7500
6cnode-set constrained
HyperPLR 0.0128 ± 0.0068 0.0198 ± 0.0042 0.0121 ± 0.0102 0.0072 ± 0.0032 1.5000
JanusNC 0.0119 ± 0.0035 0.0111 ± 0.0041 0.0154 ± 0.0024 0.0138 ± 0.0013 1.5000
Hyperlap+ 0.1821 ± 0.0488 0.1187 ± 0.0279 0.0943 ± 0.0332 0.0920 ± 0.0366 3.2500
Hyperlap 0.1571 ± 0.0195 0.1497 ± 0.0139 0.1632 ± 0.0171 0.1581 ± 0.0164 3.7500
HyperDK00 0.3818 ± 0.0050 0.3813 ± 0.0052 0.3394 ± 0.0025 0.3408 ± 0.0025 5.0000
HyperDK11 0.4252 ± 0.0057 0.4151 ± 0.0084 0.3439 ± 0.0018 0.3440 ± 0.0016 6.0000
LaTeX table source ↗
macroscopic · community contact-high-school

Macroscopic community performance comparison on the contact-high-school dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Modularity (lower is better), Hypergraph Modularity (lower is better), Bipartite Modularity (lower is better),

Model Clique Exp. Modularity Hypergraph Modularity Bipartite Modularity Line Graph Modularity Avg. Rank
6cunconstrained
Janus 0.0109 ± 0.0058 0.0450 ± 0.0034 0.0116 ± 0.0056 0.0044 ± 0.0019 1.0000
TheRA 0.1980 0.0776 0.0784 0.0783 2.0000
HyperPA 0.4910 ± 0.0023 0.5052 ± 0.0126 0.2841 ± 0.0017 0.2838 ± 0.0017 3.2500
SubsetSampling 0.4185 ± 0.0048 0.5078 ± 0.0042 0.2906 ± 0.0030 0.2905 ± 0.0030 3.7500
NaivePA 0.5214 ± 0.0020 0.5904 ± 0.0304 0.4264 ± 0.0012 0.4261 ± 0.0018 5.0000
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0009 ± 0.0002 0.0034 ± 0.0044 0.0020 ± 0.0016 0.0016 ± 0.0011 1.2500
Janus--VAE 0.0047 ± 0.0064 0.0029 ± 0.0037 0.0026 ± 0.0024 0.0027 ± 0.0013 1.7500
6cnode-set constrained
JanusNC 0.0011 ± 0.0007 0.0189 ± 0.0038 0.0005 ± 0.0004 0.0040 ± 0.0013 1.2500
Hyperlap+ 0.1608 ± 0.0016 0.0776 ± 0.0012 0.0052 ± 0.0009 0.0037 ± 0.0019 2.0000
HyperPLR 0.0271 ± 0.0016 0.1652 ± 0.0021 0.1009 ± 0.0055 0.1014 ± 0.0032 3.0000
Hyperlap 0.1924 ± 0.0134 0.1544 ± 0.0049 0.1304 ± 0.0031 0.1305 ± 0.0029 3.7500
HyperDK11 0.4983 ± 0.0034 0.5135 ± 0.0021 0.2950 ± 0.0002 0.2948 ± 0.0005 5.2500
HyperDK00 0.5154 ± 0.0014 0.5110 ± 0.0095 0.3778 ± 0.0007 0.3773 ± 0.0017 5.7500
LaTeX table source ↗
macroscopic · community contact-primary-school

Macroscopic community performance comparison on the contact-primary-school dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Modularity (lower is better), Hypergraph Modularity (lower is better), Bipartite Modularity (lower is bette

Model Clique Exp. Modularity Hypergraph Modularity Bipartite Modularity Line Graph Modularity Avg. Rank
6cunconstrained
Janus 0.0119 ± 0.0025 0.0224 ± 0.0018 0.0103 ± 0.0016 0.0109 ± 0.0024 1.0000
TheRA 0.1358 0.1713 0.1219 0.1224 2.0000
HyperPA 0.2179 ± 0.0057 0.3517 ± 0.0297 0.2008 ± 0.0032 0.2004 ± 0.0021 3.2500
SubsetSampling 0.1531 ± 0.0101 0.3656 ± 0.0126 0.2102 ± 0.0020 0.2101 ± 0.0013 3.7500
NaivePA 0.2409 ± 0.0020 0.3951 ± 0.0142 0.3309 ± 0.0009 0.3308 ± 0.0010 5.0000
6c(Janus) reconstruction upper bound
Janus--VAE 0.0010 ± 0.0003 0.0002 ± 0.0001 0.0007 ± 0.0006 0.0016 ± 0.0012 1.5000
JanusNC--VAE 0.0001 ± 0.0001 0.0002 ± 0.0002 0.0010 ± 0.0009 0.0008 ± 0.0006 1.5000
6cnode-set constrained
JanusNC 0.0058 ± 0.0030 0.0123 ± 0.0009 0.0023 ± 0.0007 0.0031 ± 0.0030 1.5000
Hyperlap 0.0034 ± 0.0030 0.0549 ± 0.0029 0.0156 ± 0.0019 0.0165 ± 0.0019 2.0000
HyperPLR 0.0026 ± 0.0009 0.0603 ± 0.0031 0.0352 ± 0.0018 0.0352 ± 0.0015 3.0000
Hyperlap+ 0.2123 ± 0.0006 0.0866 ± 0.0083 0.0207 ± 0.0141 0.0211 ± 0.0142 3.5000
HyperDK11 0.2402 ± 0.0012 0.3610 ± 0.0127 0.2159 ± 0.0003 0.2155 ± 0.0003 5.0000
HyperDK00 0.2538 ± 0.0010 0.3686 ± 0.0256 0.2961 ± 0.0008 0.2958 ± 0.0005 6.0000
LaTeX table source ↗
macroscopic · community email-Enron

Macroscopic community performance comparison on the email-Enron dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Modularity (lower is better), Hypergraph Modularity (lower is better), Bipartite Modularity (lower is better), Line Gr

Model Clique Exp. Modularity Hypergraph Modularity Bipartite Modularity Line Graph Modularity Avg. Rank
6cunconstrained
Janus 0.0087 ± 0.0027 0.0047 ± 0.0039 0.0214 ± 0.0076 0.0235 ± 0.0106 1.0000
TheRA 0.1929 0.1377 0.0779 0.0734 2.2500
HyperPA 0.2297 ± 0.0136 0.3770 ± 0.0149 0.2626 ± 0.0138 0.2575 ± 0.0155 3.2500
SubsetSampling 0.0637 ± 0.0303 0.3964 ± 0.0411 0.2690 ± 0.0065 0.2668 ± 0.0060 3.5000
NaivePA 0.2737 ± 0.0051 0.4661 ± 0.0343 0.3612 ± 0.0059 0.3596 ± 0.0063 5.0000
6c(Janus) reconstruction upper bound
Janus--VAE 0.0096 ± 0.0072 0.0006 ± 0.0003 0.0014 ± 0.0009 0.0030 ± 0.0033 1.5000
JanusNC--VAE 0.0068 ± 0.0036 0.0003 ± 0.0002 0.0050 ± 0.0046 0.0068 ± 0.0032 1.5000
6cnode-set constrained
JanusNC 0.0067 ± 0.0035 0.0042 ± 0.0018 0.0060 ± 0.0025 0.0069 ± 0.0045 1.2500
Hyperlap 0.0693 ± 0.0459 0.0246 ± 0.0065 0.0808 ± 0.0126 0.0775 ± 0.0117 2.2500
HyperPLR 0.0060 ± 0.0035 0.0794 ± 0.0109 0.0901 ± 0.0035 0.0852 ± 0.0041 3.0000
Hyperlap+ 0.2379 ± 0.0085 0.2047 ± 0.0010 0.0817 ± 0.0013 0.0846 ± 0.0013 3.5000
HyperDK11 0.2666 ± 0.0025 0.3731 ± 0.0044 0.2771 ± 0.0010 0.2755 ± 0.0021 5.0000
HyperDK00 0.2724 ± 0.0027 0.4176 ± 0.0474 0.2969 ± 0.0021 0.2950 ± 0.0021 6.0000
LaTeX table source ↗
macroscopic · community email-Eu

Macroscopic community performance comparison on the email-Eu dataset. Methods are grouped by kind. Reported metrics: Clique Exp. Modularity (lower is better), Hypergraph Modularity (lower is better), Bipartite Modularity (lower is better), Line Graph

Model Clique Exp. Modularity Hypergraph Modularity Bipartite Modularity Line Graph Modularity Avg. Rank
6cunconstrained
Janus 0.1156 ± 0.0076 0.0174 ± 0.0044 0.1060 ± 0.0017 0.1067 ± 0.0020 1.0000
TheRA 0.2714 0.2836 0.1133 0.1169 2.0000
SubsetSampling 0.3097 ± 0.0031 0.4463 ± 0.0258 0.3493 ± 0.0034 0.3507 ± 0.0035 3.2500
HyperPA 0.2992 ± 0.0036 0.5012 ± 0.0315 0.3629 ± 0.0013 0.3641 ± 0.0014 3.7500
NaivePA 0.3132 ± 0.0051 0.5091 ± 0.0077 0.4267 ± 0.0017 0.4275 ± 0.0018 5.0000
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0019 ± 0.0007 0.0017 ± 0.0014 0.0031 ± 0.0014 0.0022 ± 0.0007 1.2500
Janus--VAE 0.0089 ± 0.0046 0.0012 ± 0.0008 0.0135 ± 0.0077 0.0033 ± 0.0024 1.7500
6cnode-set constrained
JanusNC 0.0193 ± 0.0042 0.0667 ± 0.0048 0.0397 ± 0.0029 0.0459 ± 0.0038 1.2500
Hyperlap 0.0924 ± 0.0091 0.0067 ± 0.0013 0.1079 ± 0.0043 0.1084 ± 0.0042 2.5000
HyperPLR 0.0569 ± 0.0044 0.1864 ± 0.0085 0.1863 ± 0.0033 0.1869 ± 0.0048 3.2500
Hyperlap+ 0.3728 ± 0.0012 0.2249 ± 0.0020 0.0686 ± 0.0042 0.0727 ± 0.0030 3.5000
HyperDK00 0.3260 ± 0.0006 0.4256 ± 0.0056 0.3768 ± 0.0006 0.3777 ± 0.0007 5.2500
HyperDK11 0.3327 ± 0.0012 0.4420 ± 0.0083 0.3635 ± 0.0002 0.3649 ± 0.0004 5.2500
macroscopic · structure NDC-classes

Macroscopic structure performance comparison on the NDC-classes dataset. Methods are grouped by kind. Reported metrics: Incidence Matrix Density (lower is better), LCC Diameter (lower is better), Clique Exp. # Edges (lower is better), Line Graph # Ed

Model Incidence Matrix Density LCC Diameter Clique Exp. # Edges Line Graph # Edges Avg. Rank
6cunconstrained
Janus 0.0046 ± 0.0006 0.8000 ± 0.4472 159.0000 ± 142.0194 82433.8000 ± 15422.4404 1.5000
TheRA 0.0004 2.0000 10166.0000 25451.0000 2.2500
SubsetSampling 0.0046 ± 0.0006 4.0000 ± 0.0000 4380.8000 ± 646.1391 97821.0000 ± 40655.9604 3.0000
HyperPA 0.0049 ± 0.0003 5.0000 ± 0.0000 1851.4000 ± 367.2285 214178.8000 ± 62837.8174 3.8750
NaivePA 0.0063 ± 0.0002 5.0000 ± 0.0000 5856.8000 ± 914.3502 180947.8000 ± 53300.6257 4.3750
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 0.0000 ± 0.0000 5.4000 ± 5.5946 375.4000 ± 380.2714 1.0000
Janus--VAE 0.0000 ± 0.0000 0.4000 ± 0.8944 54.2000 ± 22.5765 2107.4000 ± 279.9898 2.0000
6cnode-set constrained
JanusNC 0.0012 ± 0.0001 0.2000 ± 0.4472 98.6000 ± 6.5803 11413.0000 ± 751.0310 2.2500
HyperPLR 0.0010 ± 0.0002 0.0000 ± 0.0000 706.4000 ± 7.4364 34292.2000 ± 4907.9905 2.5000
Hyperlap+ 0.0014 ± 0.0000 2.6000 ± 0.5477 8998.6000 ± 280.6533 9905.6000 ± 3257.1133 3.2500
HyperDK00 0.0000 ± 0.0000 4.6000 ± 0.5477 12545.0000 ± 69.6671 18179.0000 ± 258.0949 3.7500
HyperDK11 0.0001 ± 0.0000 4.2000 ± 0.4472 12290.6000 ± 143.4932 71341.2000 ± 916.8335 4.2500
Hyperlap 0.0013 ± 0.0001 3.8000 ± 0.4472 13230.2000 ± 291.4107 35283.8000 ± 837.5964 5.0000
LaTeX table source ↗
macroscopic · structure contact-high-school

Macroscopic structure performance comparison on the contact-high-school dataset. Methods are grouped by kind. Reported metrics: Incidence Matrix Density (lower is better), LCC Diameter (lower is better), Clique Exp. # Edges (lower is better), Line Gr

Model Incidence Matrix Density LCC Diameter Clique Exp. # Edges Line Graph # Edges Avg. Rank
6cunconstrained
Janus 0.0000 ± 0.0000 0.8000 ± 0.4472 391.6000 ± 35.1184 24882.0000 ± 5694.4898 1.8750
HyperPA 0.0002 ± 0.0001 0.0000 ± 0.0000 616.4000 ± 248.7877 734993.2000 ± 70823.9485 2.7500
SubsetSampling 0.0001 ± 0.0001 0.2000 ± 0.4472 1100.0000 ± 220.3497 540273.2000 ± 73085.8941 2.7500
TheRA 0.0009 1.0000 603.0000 12563.0000 3.0000
NaivePA 0.0033 ± 0.0001 0.8000 ± 0.4472 6396.4000 ± 442.5306 2613255.8000 ± 334047.7548 4.6250
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 0.0000 ± 0.0000 2.8000 ± 2.1679 2346.2000 ± 439.4271 1.1250
Janus--VAE 0.0000 ± 0.0000 0.0000 ± 0.0000 21.0000 ± 2.0000 10152.4000 ± 1683.2713 1.8750
6cnode-set constrained
JanusNC 0.0001 ± 0.0000 0.0000 ± 0.0000 141.0000 ± 6.8191 4666.0000 ± 502.5221 1.7500
Hyperlap+ 0.0000 ± 0.0000 1.0000 ± 0.0000 1218.8000 ± 43.8030 13909.2000 ± 2474.6313 2.6250
Hyperlap 0.0000 ± 0.0000 0.8000 ± 0.4472 2865.2000 ± 49.1650 16890.4000 ± 2438.8031 2.8750
HyperDK11 0.0000 ± 0.0000 0.8000 ± 0.4472 5217.6000 ± 29.7456 42017.2000 ± 1479.0841 3.6250
HyperPLR 0.0012 ± 0.0000 1.0000 ± 0.0000 829.6000 ± 5.6833 338960.4000 ± 5610.0259 4.3750
HyperDK00 0.0020 ± 0.0000 2.0000 ± 0.0000 11568.6000 ± 114.3451 176494.8000 ± 2118.9379 5.7500
LaTeX table source ↗
macroscopic · structure contact-primary-school

Macroscopic structure performance comparison on the contact-primary-school dataset. Methods are grouped by kind. Reported metrics: Incidence Matrix Density (lower is better), LCC Diameter (lower is better), Clique Exp. # Edges (lower is better), Line

Model Incidence Matrix Density LCC Diameter Clique Exp. # Edges Line Graph # Edges Avg. Rank
6cunconstrained
Janus 0.0000 ± 0.0000 0.2000 ± 0.4472 339.8000 ± 22.6208 79106.8000 ± 8182.4088 1.8750
TheRA 0.0014 0.0000 714.0000 326828.0000 2.7500
HyperPA 0.0001 ± 0.0000 0.2000 ± 0.4472 580.0000 ± 419.0692 1606982.2000 ± 443553.1759 2.8750
SubsetSampling 0.0001 ± 0.0000 0.0000 ± 0.0000 829.4000 ± 474.7924 1985241.6000 ± 784304.1783 3.2500
NaivePA 0.0042 ± 0.0000 0.0000 ± 0.0000 4214.4000 ± 624.7234 7413493.4000 ± 1293515.7968 4.2500
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 0.0000 ± 0.0000 2.0000 ± 1.4142 6485.8000 ± 2344.5962 1.1250
Janus--VAE 0.0000 ± 0.0000 0.0000 ± 0.0000 26.8000 ± 5.0200 34578.2000 ± 2824.2333 1.8750
6cnode-set constrained
JanusNC 0.0000 ± 0.0000 0.0000 ± 0.0000 143.8000 ± 12.1943 11737.0000 ± 1662.6089 1.8750
Hyperlap 0.0000 ± 0.0000 0.2000 ± 0.4472 1934.0000 ± 62.1410 7266.8000 ± 3264.9594 2.3750
Hyperlap+ 0.0000 ± 0.0000 1.0000 ± 0.0000 263.4000 ± 214.5001 67957.6000 ± 21124.7408 2.8750
HyperPLR 0.0010 ± 0.0000 0.0000 ± 0.0000 1455.0000 ± 0.0000 622811.4000 ± 31925.1994 3.6250
HyperDK11 0.0001 ± 0.0000 1.0000 ± 0.0000 6388.8000 ± 29.3462 357168.2000 ± 9569.1507 4.5000
HyperDK00 0.0029 ± 0.0000 1.0000 ± 0.0000 12522.0000 ± 77.9519 676662.8000 ± 5594.9212 5.7500
LaTeX table source ↗
macroscopic · structure email-Enron

Macroscopic structure performance comparison on the email-Enron dataset. Methods are grouped by kind. Reported metrics: Incidence Matrix Density (lower is better), LCC Diameter (lower is better), Clique Exp. # Edges (lower is better), Line Graph # Ed

Model Incidence Matrix Density LCC Diameter Clique Exp. # Edges Line Graph # Edges Avg. Rank
6cunconstrained
Janus 0.0021 ± 0.0003 0.0000 ± 0.0000 13.2000 ± 10.5451 20514.4000 ± 3063.9399 1.8750
HyperPA 0.0018 ± 0.0010 0.0000 ± 0.0000 244.4000 ± 154.3869 31887.8000 ± 26026.0117 2.1250
SubsetSampling 0.0017 ± 0.0006 0.2000 ± 0.4472 431.0000 ± 107.2357 41643.8000 ± 11717.4138 2.7500
TheRA 0.0026 1.0000 1275.0000 14821.0000 3.6250
NaivePA 0.0098 ± 0.0008 1.0000 ± 0.0000 1038.8000 ± 178.2939 112411.2000 ± 54623.5179 4.6250
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 0.0000 ± 0.0000 2.0000 ± 4.4721 266.4000 ± 332.3316 1.2500
Janus--VAE 0.0000 ± 0.0000 0.0000 ± 0.0000 2.0000 ± 2.1213 461.8000 ± 388.9758 1.7500
6cnode-set constrained
JanusNC 0.0003 ± 0.0001 0.0000 ± 0.0000 15.0000 ± 4.6368 2228.2000 ± 821.6256 1.3750
Hyperlap+ 0.0009 ± 0.0000 0.2000 ± 0.4472 686.0000 ± 35.3483 3603.8000 ± 1964.1286 3.0000
HyperDK11 0.0001 ± 0.0001 1.0000 ± 0.0000 2161.2000 ± 22.6539 5507.4000 ± 878.6420 3.6250
HyperPLR 0.0020 ± 0.0002 0.0000 ± 0.0000 62.4000 ± 3.3615 23527.8000 ± 3819.9611 3.6250
Hyperlap 0.0009 ± 0.0001 1.0000 ± 0.0000 1846.4000 ± 106.7183 2849.4000 ± 1528.9669 3.6250
HyperDK00 0.0020 ± 0.0001 2.0000 ± 0.0000 3064.8000 ± 28.3231 22566.4000 ± 422.1194 5.7500
LaTeX table source ↗
macroscopic · structure email-Eu

Macroscopic structure performance comparison on the email-Eu dataset. Methods are grouped by kind. Reported metrics: Incidence Matrix Density (lower is better), LCC Diameter (lower is better), Clique Exp. # Edges (lower is better), Line Graph # Edges

Model Incidence Matrix Density LCC Diameter Clique Exp. # Edges Line Graph # Edges Avg. Rank
6cunconstrained
SubsetSampling 0.0000 ± 0.0000 2.0000 ± 0.0000 28833.4000 ± 3498.3531 10460547.0000 ± 2901237.0644 2.0000
Janus 0.0023 ± 0.0000 0.6667 ± 0.5774 16744.6667 ± 248.4056 22185259.0000 ± 258695.8816 2.7500
HyperPA 0.0003 ± 0.0003 2.2000 ± 0.4472 10587.2000 ± 1468.2138 34082994.6000 ± 3768618.3128 2.8750
TheRA 0.0013 4.0000 32109.0000 356438.0000 3.5000
NaivePA 0.0010 ± 0.0000 2.2000 ± 0.4472 36980.0000 ± 1142.5067 30468155.6000 ± 5423212.4123 3.8750
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 0.0000 ± 0.0000 57.2000 ± 19.1233 123360.2000 ± 18285.6379 1.1250
Janus--VAE 0.0001 ± 0.0000 0.0000 ± 0.0000 755.0000 ± 69.9393 700513.2000 ± 31178.1844 1.8750
6cnode-set constrained
HyperPLR 0.0002 ± 0.0000 0.2000 ± 0.4472 13615.0000 ± 88.9326 1239056.4000 ± 243479.8529 2.2500
Hyperlap+ 0.0004 ± 0.0000 0.8000 ± 0.4472 20184.4000 ± 306.8897 165310.4000 ± 28866.5000 3.0000
HyperDK11 0.0001 ± 0.0000 2.0000 ± 0.0000 69432.0000 ± 284.6621 470975.4000 ± 31351.5873 3.1250
JanusNC 0.0009 ± 0.0000 0.4000 ± 0.5477 2618.0000 ± 155.0048 3887615.2000 ± 94701.3252 3.5000
Hyperlap 0.0004 ± 0.0000 2.0000 ± 0.0000 57953.8000 ± 1494.2255 769716.6000 ± 85962.1059 3.8750
HyperDK00 0.0003 ± 0.0000 4.0000 ± 0.0000 97434.4000 ± 438.4140 4874286.4000 ± 3305.5184 5.2500
macroscopic · hypertrans overlapness contact-high-school

HyperTrans and Overlapness performance comparison on the contact-high-school dataset. Methods are grouped by kind. Reported metrics: HyperTrans (lower is better), Global Overlapness (lower is better), Egonet Overlapness (lower is better). Colored cel

Model HyperTrans Global Overlapness Egonet Overlapness Avg. Rank
5cunconstrained
Janus 0.0280 ± 0.0027 0.2615 ± 0.0763 0.1879 ± 0.0152 1.3333
HyperPA 0.2821 ± 0.0056 2.0974 ± 1.8201 0.9480 ± 0.0328 3.0000
TheRA 0.0014 6.7492 2.0610 3.0000
SubsetSampling 0.1073 ± 0.0134 8.0551 ± 1.1319 1.3749 ± 0.0112 3.6667
NaivePA 0.2527 ± 0.0038 19.8167 ± 2.7867 1.1368 ± 0.0524 4.0000
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0005 ± 0.0001 0.1058 ± 0.0195 0.0061 ± 0.0018 1.0000
Janus--VAE 0.0023 ± 0.0004 0.4606 ± 0.0672 0.0282 ± 0.0060 2.0000
5cnode-set constrained
Hyperlap+ 0.0249 ± 0.0015 0.0000 ± 0.0000 0.6173 ± 0.0178 1.6667
JanusNC 0.0075 ± 0.0010 0.5020 ± 0.0918 0.0302 ± 0.0040 1.6667
Hyperlap 0.1665 ± 0.0018 0.0683 ± 0.0935 1.1299 ± 0.0128 2.6667
HyperDK11 0.2352 ± 0.0010 2.5694 ± 0.0745 1.6703 ± 0.0046 4.6667
HyperPLR 0.2109 ± 0.0029 9.3780 ± 0.0729 1.6658 ± 0.0168 4.6667
HyperDK00 0.3173 ± 0.0007 4.8538 ± 0.1029 1.8636 ± 0.0063 5.6667
LaTeX table source ↗
macroscopic · hypertrans overlapness contact-primary-school

HyperTrans and Overlapness performance comparison on the contact-primary-school dataset. Methods are grouped by kind. Reported metrics: HyperTrans (lower is better), Global Overlapness (lower is better), Egonet Overlapness (lower is better). Colored

Model HyperTrans Global Overlapness Egonet Overlapness Avg. Rank
5cunconstrained
Janus 0.0136 ± 0.0023 0.3316 ± 0.1783 0.1265 ± 0.0211 1.3333
HyperPA 0.1999 ± 0.0038 10.1188 ± 6.1790 1.0053 ± 0.0948 3.0000
TheRA 0.0024 18.2727 2.7430 3.0000
SubsetSampling 0.0202 ± 0.0050 22.2888 ± 7.6265 1.4931 ± 0.0780 3.6667
NaivePA 0.2036 ± 0.0085 34.7454 ± 8.3622 0.5792 ± 0.1201 4.0000
5c(Janus) reconstruction upper bound
JanusNC--VAE 0.0002 ± 0.0001 0.1810 ± 0.0486 0.0064 ± 0.0011 1.0000
Janus--VAE 0.0018 ± 0.0002 1.0091 ± 0.0713 0.0346 ± 0.0049 2.0000
5cnode-set constrained
JanusNC 0.0091 ± 0.0006 0.5702 ± 0.0316 0.0537 ± 0.0070 1.6667
Hyperlap+ 0.0119 ± 0.0088 0.0000 ± 0.0000 0.1621 ± 0.1541 1.8333
Hyperlap 0.0558 ± 0.0019 0.0000 ± 0.0000 0.9482 ± 0.0193 2.5000
HyperDK11 0.1293 ± 0.0012 10.7215 ± 0.2688 2.1957 ± 0.0035 4.6667
HyperPLR 0.1316 ± 0.0019 12.6132 ± 0.2169 1.9294 ± 0.0252 5.0000
HyperDK00 0.2671 ± 0.0006 12.5322 ± 0.1986 2.1645 ± 0.0062 5.3333
LaTeX table source ↗
macroscopic · size NDC-classes

Macroscopic size performance comparison on the NDC-classes dataset. Methods are grouped by kind. Reported metrics: # Nodes (lower is better), # Hyperedges (lower is better), LCC # Nodes (lower is better), LCC # Hyperedges (lower is better). Colored c

Model # Nodes # Hyperedges LCC # Nodes LCC # Hyperedges Avg. Rank
6cunconstrained
Janus 442.2000 ± 20.4377 0.0000 ± 0.0000 20.8000 ± 10.3296 193.2000 ± 9.8336 1.2500
SubsetSampling 554.4000 ± 21.1258 83.0000 ± 46.5403 26.2000 ± 21.8792 229.8000 ± 95.3950 2.7500
TheRA 12.0000 60.0000 495.0000 306.0000 3.2500
NaivePA 560.4000 ± 17.7285 82.6000 ± 24.6840 27.4000 ± 17.7285 249.4000 ± 92.1916 3.5000
HyperPA 568.2000 ± 18.9526 92.8000 ± 67.9389 50.4000 ± 15.5338 246.8000 ± 118.1194 4.2500
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 4.2000 ± 2.3875 2.0000 ± 3.9370 5.0000 ± 4.0620 1.1250
Janus--VAE 0.0000 ± 0.0000 25.0000 ± 4.8477 35.2000 ± 39.2326 37.4000 ± 20.0823 1.8750
6cnode-set constrained
JanusNC 139.4000 ± 3.5071 0.0000 ± 0.0000 20.0000 ± 4.3589 51.8000 ± 3.4928 1.8750
HyperPLR 261.4000 ± 11.0136 0.0000 ± 0.0000 19.4000 ± 2.0736 126.0000 ± 10.3682 2.6250
HyperDK11 0.0000 ± 0.0000 10.6000 ± 2.7019 527.6000 ± 3.5777 256.8000 ± 3.7014 3.8750
HyperDK00 5.4000 ± 3.3615 2.0000 ± 1.5811 527.6000 ± 3.3615 270.0000 ± 1.5811 4.1250
Hyperlap 198.0000 ± 13.2853 39.0000 ± 0.0000 335.0000 ± 13.2853 233.0000 ± 0.0000 4.2500
Hyperlap+ 205.2000 ± 7.1903 39.0000 ± 0.0000 327.8000 ± 7.1903 233.0000 ± 0.0000 4.2500
macroscopic · size contact-high-school

Macroscopic size performance comparison on the contact-high-school dataset. Methods are grouped by kind. Reported metrics: # Nodes (lower is better), # Hyperedges (lower is better), LCC # Nodes (lower is better), LCC # Hyperedges (lower is better). C

Model # Nodes # Hyperedges LCC # Nodes LCC # Hyperedges Avg. Rank
6cunconstrained
TheRA 0.0000 0.0000 0.0000 0.0000 1.1250
Janus 0.2000 ± 0.4472 0.0000 ± 0.0000 1.0000 ± 1.0000 0.8000 ± 0.8367 1.8750
HyperPA 8.2000 ± 3.7683 342.4000 ± 309.4217 8.2000 ± 3.7683 342.4000 ± 309.4217 4.0000
NaivePA 6.8000 ± 2.6833 554.6000 ± 259.1135 6.8000 ± 2.6833 554.6000 ± 259.1135 4.0000
SubsetSampling 4.4000 ± 1.8166 1241.8000 ± 118.7800 4.4000 ± 1.8166 1241.8000 ± 118.7800 4.0000
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 16.0000 ± 3.0822 0.0000 ± 0.0000 16.0000 ± 3.0822 1.2500
Janus--VAE 0.0000 ± 0.0000 46.8000 ± 4.4944 0.0000 ± 0.0000 46.8000 ± 4.4944 1.7500
6cnode-set constrained
Hyperlap+ 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 2.2500
JanusNC 0.2000 ± 0.4472 0.0000 ± 0.0000 0.2000 ± 0.4472 0.0000 ± 0.0000 3.2500
HyperDK11 0.0000 ± 0.0000 395.4000 ± 11.0589 0.0000 ± 0.0000 395.4000 ± 11.0589 3.5000
Hyperlap 0.4000 ± 0.5477 0.0000 ± 0.0000 0.4000 ± 0.5477 0.0000 ± 0.0000 3.7500
HyperDK00 0.0000 ± 0.0000 2257.6000 ± 37.7995 0.0000 ± 0.0000 2257.6000 ± 37.7995 4.0000
HyperPLR 2.0000 ± 0.0000 0.0000 ± 0.0000 2.0000 ± 0.0000 0.0000 ± 0.0000 4.2500
LaTeX table source ↗
macroscopic · size contact-primary-school

Macroscopic size performance comparison on the contact-primary-school dataset. Methods are grouped by kind. Reported metrics: # Nodes (lower is better), # Hyperedges (lower is better), LCC # Nodes (lower is better), LCC # Hyperedges (lower is better)

Model # Nodes # Hyperedges LCC # Nodes LCC # Hyperedges Avg. Rank
6cunconstrained
TheRA 0.0000 0.0000 0.0000 0.0000 1.2500
Janus 0.2000 ± 0.4472 0.0000 ± 0.0000 0.2000 ± 0.4472 0.0000 ± 0.0000 1.7500
HyperPA 1.8000 ± 0.8367 1033.0000 ± 694.5265 1.8000 ± 0.8367 1033.0000 ± 694.5265 3.5000
NaivePA 1.4000 ± 0.8944 1314.0000 ± 586.8893 1.4000 ± 0.8944 1314.0000 ± 586.8893 3.5000
SubsetSampling 2.6000 ± 1.3416 2360.6000 ± 758.3576 2.6000 ± 1.3416 2360.6000 ± 758.3576 5.0000
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 18.4000 ± 5.3666 0.0000 ± 0.0000 18.4000 ± 5.3666 1.2500
Janus--VAE 0.0000 ± 0.0000 80.0000 ± 5.8737 0.0000 ± 0.0000 80.0000 ± 5.8737 1.7500
6cnode-set constrained
Hyperlap 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 2.7500
Hyperlap+ 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 2.7500
JanusNC 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 2.7500
HyperDK11 0.0000 ± 0.0000 1241.8000 ± 32.8283 0.0000 ± 0.0000 1241.8000 ± 32.8283 4.0000
HyperPLR 1.0000 ± 0.0000 0.0000 ± 0.0000 1.0000 ± 0.0000 0.0000 ± 0.0000 4.2500
HyperDK00 0.0000 ± 0.0000 3809.2000 ± 31.6259 0.0000 ± 0.0000 3809.2000 ± 31.6259 4.5000
LaTeX table source ↗
macroscopic · size email-Enron

Macroscopic size performance comparison on the email-Enron dataset. Methods are grouped by kind. Reported metrics: # Nodes (lower is better), # Hyperedges (lower is better), LCC # Nodes (lower is better), LCC # Hyperedges (lower is better). Colored c

Model # Nodes # Hyperedges LCC # Nodes LCC # Hyperedges Avg. Rank
6cunconstrained
Janus 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 0.0000 ± 0.0000 1.2500
TheRA 0.0000 53.0000 0.0000 53.0000 1.7500
HyperPA 13.0000 ± 3.3912 253.0000 ± 116.8696 13.6000 ± 2.8810 253.6000 ± 116.1671 3.7500
SubsetSampling 8.8000 ± 2.1679 380.6000 ± 72.6726 9.0000 ± 2.2361 380.8000 ± 72.6684 4.0000
NaivePA 13.2000 ± 2.1679 316.0000 ± 114.5949 13.2000 ± 2.1679 316.0000 ± 114.5949 4.2500
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.0000 ± 0.0000 1.0000 ± 1.2247 0.0000 ± 0.0000 1.0000 ± 1.2247 1.2500
Janus--VAE 0.0000 ± 0.0000 3.0000 ± 2.0000 0.0000 ± 0.0000 3.0000 ± 2.0000 1.7500
6cnode-set constrained
Hyperlap+ 0.0000 ± 0.0000 53.0000 ± 0.0000 0.0000 ± 0.0000 53.0000 ± 0.0000 2.7500
JanusNC 0.6000 ± 0.5477 0.0000 ± 0.0000 0.6000 ± 0.5477 0.0000 ± 0.0000 3.2500
HyperDK11 0.0000 ± 0.0000 82.6000 ± 8.2946 0.0000 ± 0.0000 82.6000 ± 8.2946 3.5000
HyperPLR 2.0000 ± 0.0000 0.0000 ± 0.0000 2.0000 ± 0.0000 0.0000 ± 0.0000 3.7500
Hyperlap 0.4000 ± 0.5477 53.0000 ± 0.0000 0.4000 ± 0.5477 53.0000 ± 0.0000 3.7500
HyperDK00 0.0000 ± 0.0000 183.6000 ± 11.3490 0.0000 ± 0.0000 183.6000 ± 11.3490 4.0000
macroscopic · size email-Eu

Macroscopic size performance comparison on the email-Eu dataset. Methods are grouped by kind. Reported metrics: # Nodes (lower is better), # Hyperedges (lower is better), LCC # Nodes (lower is better), LCC # Hyperedges (lower is better). Colored cell

Model # Nodes # Hyperedges LCC # Nodes LCC # Hyperedges Avg. Rank
6cunconstrained
Janus 12.3333 ± 7.0238 0.0000 ± 0.0000 6.3333 ± 6.5064 10.0000 ± 3.6056 1.7500
TheRA 12.0000 507.0000 7.0000 488.0000 2.2500
SubsetSampling 3.6000 ± 1.3416 1356.8000 ± 823.6171 14.6000 ± 2.0736 1338.6000 ± 823.6785 2.5000
NaivePA 3.8000 ± 2.0494 1656.0000 ± 527.4263 15.2000 ± 2.0494 1637.0000 ± 527.4263 3.5000
HyperPA 80.8000 ± 79.2950 2039.4000 ± 1234.3607 71.8000 ± 69.7044 2022.4000 ± 1233.1870 5.0000
6c(Janus) reconstruction upper bound
JanusNC--VAE 0.2000 ± 0.4472 89.4000 ± 10.8995 0.0000 ± 0.0000 89.6000 ± 10.9453 1.2500
Janus--VAE 0.0000 ± 0.0000 492.6000 ± 20.5256 0.4000 ± 0.5477 493.0000 ± 20.5791 1.7500
6cnode-set constrained
JanusNC 47.8000 ± 4.0249 0.0000 ± 0.0000 35.8000 ± 5.4037 12.0000 ± 2.9155 2.8750
HyperDK11 0.0000 ± 0.0000 1378.4000 ± 43.1080 18.2000 ± 0.4472 1360.2000 ± 42.8567 3.1250
Hyperlap 46.8000 ± 4.6043 507.0000 ± 0.0000 27.8000 ± 4.6043 488.0000 ± 0.0000 3.2500
HyperDK00 0.0000 ± 0.0000 2712.4000 ± 93.5217 19.0000 ± 0.0000 2693.4000 ± 93.5217 3.8750
HyperPLR 137.4000 ± 1.5166 0.0000 ± 0.0000 118.4000 ± 1.5166 19.0000 ± 0.0000 3.8750
Hyperlap+ 49.8000 ± 4.5497 507.0000 ± 0.0000 30.8000 ± 4.5497 488.0000 ± 0.0000 4.0000
higher order dissimilarity measures NDC-classes

Higher-order dissimilarity measures performance comparison on the NDC-classes dataset. Methods are grouped by kind. Reported metrics: Hyper NetSimile (lower is better), Hyperedge Portrait Divergence (lower is better). Colored cells mark the top three

Model Hyper NetSimile Hyperedge Portrait Divergence Avg. Rank
4cunconstrained
Janus 6.0414 ± 0.3021 0.4210 ± 0.0484 1.5000
TheRA 9.7695 0.3320 2.0000
HyperPA 8.0762 ± 0.4667 0.6145 ± 0.0348 3.0000
SubsetSampling 9.8929 ± 0.4720 0.5203 ± 0.0553 3.5000
NaivePA 10.2800 ± 0.3795 0.7330 ± 0.0039 5.0000
4c(Janus) reconstruction upper bound
JanusNC--VAE 0.4208 ± 0.5567 0.0217 ± 0.0295 1.0000
Janus--VAE 1.1361 ± 0.0679 0.0561 ± 0.0299 2.0000
4cnode-set constrained
JanusNC 2.5608 ± 0.0646 0.1518 ± 0.0259 1.0000
HyperPLR 3.3487 ± 0.1112 0.2818 ± 0.0394 2.0000
HyperDK11 8.5485 ± 0.0541 0.6143 ± 0.0151 3.5000
Hyperlap+ 8.4143 ± 0.3084 0.6148 ± 0.0223 3.5000
Hyperlap 9.8272 ± 0.0645 0.6455 ± 0.0306 5.0000
HyperDK00 10.2463 ± 0.0391 0.7673 ± 0.0258 6.0000
LaTeX table source ↗
higher order dissimilarity measures contact-high-school

Higher-order dissimilarity measures performance comparison on the contact-high-school dataset. Methods are grouped by kind. Reported metrics: Hyper NetSimile (lower is better), Hyperedge Portrait Divergence (lower is better). Colored cells mark the t

Model Hyper NetSimile Hyperedge Portrait Divergence Avg. Rank
4cunconstrained
Janus 0.9456 ± 0.1792 0.3405 ± 0.0034 1.5000
TheRA 2.8572 0.3232 1.5000
HyperPA 5.2400 ± 0.0917 0.6133 ± 0.0137 3.0000
SubsetSampling 6.4425 ± 0.1279 0.8202 ± 0.0198 4.0000
NaivePA 6.5915 ± 0.1751 0.8949 ± 0.0011 5.0000
4c(Janus) reconstruction upper bound
JanusNC--VAE 0.0130 ± 0.0040 0.0738 ± 0.0165 1.0000
Janus--VAE 0.0720 ± 0.0199 0.1026 ± 0.0067 2.0000
4cnode-set constrained
JanusNC 0.1946 ± 0.0177 0.2280 ± 0.0329 1.0000
Hyperlap 2.9377 ± 0.0784 0.3532 ± 0.0045 2.5000
HyperPLR 2.0690 ± 0.0146 0.7258 ± 0.0012 3.5000
Hyperlap+ 3.0168 ± 0.0883 0.5113 ± 0.0209 3.5000
HyperDK11 3.3553 ± 0.0422 0.5258 ± 0.0135 4.5000
HyperDK00 6.0711 ± 0.0307 0.8222 ± 0.0048 6.0000
LaTeX table source ↗
higher order dissimilarity measures contact-primary-school

Higher-order dissimilarity measures performance comparison on the contact-primary-school dataset. Methods are grouped by kind. Reported metrics: Hyper NetSimile (lower is better), Hyperedge Portrait Divergence (lower is better). Colored cells mark th

Model Hyper NetSimile Hyperedge Portrait Divergence Avg. Rank
4cunconstrained
Janus 0.3685 ± 0.0322 0.2277 ± 0.0043 1.0000
HyperPA 4.0878 ± 0.1772 0.5031 ± 0.0319 2.5000
TheRA 2.0996 0.5405 2.5000
NaivePA 5.2683 ± 0.1044 0.8612 ± 0.0009 4.5000
SubsetSampling 5.8668 ± 0.0935 0.8108 ± 0.0121 4.5000
4c(Janus) reconstruction upper bound
JanusNC--VAE 0.0128 ± 0.0028 0.0876 ± 0.0176 1.0000
Janus--VAE 0.0661 ± 0.0144 0.1326 ± 0.0081 2.0000
4cnode-set constrained
JanusNC 0.1764 ± 0.0171 0.2020 ± 0.0031 1.0000
Hyperlap 2.1658 ± 0.0354 0.2114 ± 0.0031 2.5000
HyperPLR 1.6567 ± 0.0515 0.5826 ± 0.0023 3.5000
HyperDK11 3.5480 ± 0.0880 0.4578 ± 0.0046 4.0000
Hyperlap+ 2.9876 ± 0.0155 0.5179 ± 0.0087 4.0000
HyperDK00 7.6628 ± 0.0568 0.8482 ± 0.0071 6.0000
LaTeX table source ↗
higher order dissimilarity measures email-Enron

Higher-order dissimilarity measures performance comparison on the email-Enron dataset. Methods are grouped by kind. Reported metrics: Hyper NetSimile (lower is better), Hyperedge Portrait Divergence (lower is better). Colored cells mark the top three

Model Hyper NetSimile Hyperedge Portrait Divergence Avg. Rank
4cunconstrained
Janus 0.8984 ± 0.0678 0.1158 ± 0.0086 1.0000
HyperPA 3.8839 ± 0.2596 0.2858 ± 0.0575 3.0000
TheRA 3.5654 0.5158 3.0000
NaivePA 3.8490 ± 0.4478 0.6531 ± 0.0097 4.0000
SubsetSampling 4.1982 ± 0.1851 0.4526 ± 0.0382 4.0000
4c(Janus) reconstruction upper bound
JanusNC--VAE 0.0125 ± 0.0216 0.0029 ± 0.0046 1.0000
Janus--VAE 0.0263 ± 0.0239 0.0073 ± 0.0063 2.0000
4cnode-set constrained
JanusNC 0.1391 ± 0.0181 0.0461 ± 0.0031 1.0000
HyperPLR 0.6153 ± 0.0898 0.2680 ± 0.0090 2.0000
HyperDK11 3.2826 ± 0.1244 0.4482 ± 0.0370 3.0000
Hyperlap+ 3.5464 ± 0.3430 0.5310 ± 0.0259 4.0000
Hyperlap 3.7188 ± 0.1171 0.5543 ± 0.0277 5.0000
HyperDK00 6.7569 ± 0.0927 0.7562 ± 0.0263 6.0000
LaTeX table source ↗
higher order dissimilarity measures email-Eu

Higher-order dissimilarity measures performance comparison on the email-Eu dataset. Methods are grouped by kind. Reported metrics: Hyper NetSimile (lower is better), Hyperedge Portrait Divergence (lower is better). Colored cells mark the top three va

Model Hyper NetSimile Hyperedge Portrait Divergence Avg. Rank
4cunconstrained
Janus 4.1857 ± 0.0573 0.4227 ± 0.0470 1.0000
SubsetSampling 4.6115 ± 0.3025 0.4287 ± 0.0390 2.0000
TheRA 6.1181 0.7134 3.5000
HyperPA 7.9956 ± 0.2381 0.5949 ± 0.0143 4.0000
NaivePA 6.2691 ± 0.2019 0.7350 ± 0.0033 4.5000
4c(Janus) reconstruction upper bound
JanusNC--VAE 0.0387 ± 0.0078 0.0574 ± 0.0341 1.0000
Janus--VAE 0.3138 ± 0.0268 0.1125 ± 0.0023 2.0000
4cnode-set constrained
JanusNC 1.2224 ± 0.0476 0.3009 ± 0.0305 1.0000
HyperDK11 4.4075 ± 0.0247 0.4309 ± 0.0036 2.0000
Hyperlap 4.6837 ± 0.1403 0.5742 ± 0.0388 3.0000
Hyperlap+ 4.7418 ± 0.0824 0.6210 ± 0.0114 4.0000
HyperPLR 4.8988 ± 0.0219 0.7806 ± 0.0097 5.0000
HyperDK00 9.8383 ± 0.0197 0.8432 ± 0.0032 6.0000
LaTeX table source ↗
conditional reconstruction NDC-classes

Reconstruction performance comparison on the NDC-classes dataset. Methods are grouped by kind. Reported metrics: Intersection (higher is better), Jaccard Coefficient (higher is better). Colored cells mark the top three values within each kind for ran

Model Intersection Jaccard Coefficient Avg. Rank
4c(Janus) reconstruction upper bound
JanusNC--VAE 1088.0000 ± 0.0000 0.9962 ± 0.0022 1.0000
Janus--VAE 1087.6000 ± 0.5477 0.9768 ± 0.0035 2.0000
4cnode-set constrained
JanusNC 971.8000 ± 4.9193 0.8070 ± 0.0074 1.0000
HyperPLR 348.4000 ± 10.0648 0.1907 ± 0.0065 2.0000
HyperDK11 6.4000 ± 1.8166 0.0030 ± 0.0008 3.0000
Hyperlap+ 5.0000 ± 2.3452 0.0024 ± 0.0011 4.0000
Hyperlap 3.4000 ± 1.3416 0.0016 ± 0.0006 5.0000
HyperDK00 2.2000 ± 0.8367 0.0010 ± 0.0004 6.0000
LaTeX table source ↗
conditional reconstruction contact-high-school

Reconstruction performance comparison on the contact-high-school dataset. Methods are grouped by kind. Reported metrics: Intersection (higher is better), Jaccard Coefficient (higher is better). Colored cells mark the top three values within each kind

Model Intersection Jaccard Coefficient Avg. Rank
4c(Janus) reconstruction upper bound
JanusNC--VAE 7818.0000 ± 0.0000 0.9980 ± 0.0004 1.2500
Janus--VAE 7818.0000 ± 0.0000 0.9940 ± 0.0006 1.7500
4cnode-set constrained
JanusNC 7616.8000 ± 7.3280 0.9498 ± 0.0018 1.0000
HyperPLR 2765.8000 ± 31.6654 0.2149 ± 0.0030 2.0000
HyperDK11 574.8000 ± 31.0596 0.0392 ± 0.0022 3.0000
Hyperlap 494.6000 ± 24.0687 0.0352 ± 0.0018 4.0000
Hyperlap+ 463.4000 ± 34.1877 0.0338 ± 0.0026 5.0000
HyperDK00 225.0000 ± 9.8742 0.0171 ± 0.0007 6.0000
LaTeX table source ↗
conditional reconstruction contact-primary-school

Reconstruction performance comparison on the contact-primary-school dataset. Methods are grouped by kind. Reported metrics: Intersection (higher is better), Jaccard Coefficient (higher is better). Colored cells mark the top three values within each k

Model Intersection Jaccard Coefficient Avg. Rank
4c(Janus) reconstruction upper bound
JanusNC--VAE 12704.0000 ± 0.0000 0.9986 ± 0.0004 1.2500
Janus--VAE 12704.0000 ± 0.0000 0.9937 ± 0.0005 1.7500
4cnode-set constrained
JanusNC 12243.2000 ± 20.8014 0.9300 ± 0.0030 1.0000
HyperPLR 5127.0000 ± 50.2643 0.2528 ± 0.0031 2.0000
HyperDK11 1780.8000 ± 81.0876 0.0796 ± 0.0039 3.0000
Hyperlap 1390.0000 ± 21.7141 0.0657 ± 0.0011 4.0000
Hyperlap+ 1227.2000 ± 69.4421 0.0608 ± 0.0035 5.0000
HyperDK00 857.4000 ± 14.3108 0.0413 ± 0.0007 6.0000
LaTeX table source ↗
conditional reconstruction email-Enron

Reconstruction performance comparison on the email-Enron dataset. Methods are grouped by kind. Reported metrics: Intersection (higher is better), Jaccard Coefficient (higher is better). Colored cells mark the top three values within each kind for ran

Model Intersection Jaccard Coefficient Avg. Rank
4c(Janus) reconstruction upper bound
JanusNC--VAE 1512.0000 ± 0.0000 0.9993 ± 0.0008 1.2500
Janus--VAE 1512.0000 ± 0.0000 0.9980 ± 0.0013 1.7500
4cnode-set constrained
JanusNC 1450.2000 ± 4.8683 0.9215 ± 0.0060 1.0000
HyperPLR 419.8000 ± 19.0316 0.1613 ± 0.0085 2.0000
HyperDK11 82.2000 ± 5.0695 0.0287 ± 0.0018 3.0000
HyperDK00 74.8000 ± 5.9330 0.0270 ± 0.0022 4.0000
Hyperlap 53.2000 ± 6.0581 0.0194 ± 0.0023 5.0000
Hyperlap+ 47.8000 ± 7.9812 0.0178 ± 0.0030 6.0000
LaTeX table source ↗
conditional reconstruction email-Eu

Reconstruction performance comparison on the email-Eu dataset. Methods are grouped by kind. Reported metrics: Intersection (higher is better), Jaccard Coefficient (higher is better). Colored cells mark the top three values within each kind for ranked

Model Intersection Jaccard Coefficient Avg. Rank
4c(Janus) reconstruction upper bound
JanusNC--VAE 25005.0000 ± 3.6742 0.9947 ± 0.0003 1.0000
Janus--VAE 24983.6000 ± 2.0736 0.9773 ± 0.0008 2.0000
4cnode-set constrained
JanusNC 14693.4000 ± 38.7595 0.4155 ± 0.0016 1.0000
HyperPLR 3297.2000 ± 85.4939 0.0705 ± 0.0020 2.0000
HyperDK11 1022.4000 ± 66.8566 0.0215 ± 0.0014 3.0000
HyperDK00 715.6000 ± 11.7175 0.0153 ± 0.0002 4.0000
Hyperlap 652.2000 ± 40.7087 0.0140 ± 0.0009 5.0000
Hyperlap+ 572.6000 ± 17.5585 0.0127 ± 0.0004 6.0000
LaTeX table source ↗
dgmh-comparison

Comparison results using the DGMH evaluation framework on NDC-classes and email-Eu datasets.

Dataset Metric Motif Real JanusNC DGMH Best
4*Clust. coeff.
Node 0.49 0.52 0.31 JanusNC 0.03
Edge 0.56 0.66 0.69 JanusNC 0.10
Triangle 0.82 0.80 0.83 DGMH 0.01
4-clique 0.89 0.86 0.91 DGMH 0.02
4*Clust. coeff.
Node 0.61 0.66 0.64 DGMH 0.03
Edge 0.71 0.89 0.92 JanusNC 0.18
Triangle 0.80 0.95 0.97 JanusNC 0.15
4-clique 0.74 0.95 0.97 JanusNC 0.21
4*Conn. comp.
Node 0.98 0.99 1.00 JanusNC 0.01
Edge 0.70 0.84 0.77 DGMH 0.07
Triangle 0.80 0.86 0.76 DGMH 0.04
4-clique 0.41 0.54 0.42 DGMH 0.01
4*Conn. comp.
Node 0.54 0.60 0.73 JanusNC 0.06
Edge 0.58 0.62 0.79 JanusNC 0.04
Triangle 0.26 0.29 0.23 Tie 0.03
4-clique 0.19 0.20 0.20 Tie 0.01
4*Diameter
Node 2.82 2.82 1.67 JanusNC 0.00
Edge 5.71 5.73 5.48 JanusNC 0.02
Triangle 10.40 10.63 9.51 JanusNC 0.23
4-clique 14.90 16.59 18.62 JanusNC 1.69
4*Diameter
Node 4.45 4.50 4.91 JanusNC 0.05
Edge 6.62 6.58 6.82 JanusNC 0.04
Triangle 6.51 6.28 4.68 JanusNC 0.23
4-clique 2.69 2.70 2.57 JanusNC 0.01
6r|Overall Best & JanusNC &
Ablation studies2 tables
ablation kmeans results

Aggregated clustering metrics by dataset.

Dataset ARI NMI AMI
contact-high-school 0.97774 0.99512 0.99490
contact-primary-school 0.99224 0.99816 0.99816
email-Enron 0.99788 0.99914 0.99912
email-Eu 0.99473 0.99972 0.99972
NDC-classes 0.50184 0.88380 0.86824
no-cross-attention

Comparison between Janus with and without Cross-Attention (Janus*). Improvement is reported as the relative improvement of Janus over Janus*.

Janus Janus* Improvement (%)
Incidence Matrix Density 0.00209 0.00385 45.64492
Largest Connected Component Diameter 0.00000 0.40000 100.00000
Number of Nodes 0.00000 1.40000 100.00000
Number of Hyperedges 0.00000 0.00000 0.00000
Clique Expansion Number of Edges 13.20000 50.40000 73.80952
Line Graph Number of Edges 20514.40000 35809.60000 42.71257
Hypergraph Modularity 0.00472 0.01333 64.63398
Clique Expansion Modularity 0.00868 0.00789 -10.00372
Line Graph Modularity 0.02347 0.02837 17.30312
Bipartite Graph Modularity 0.02136 0.02845 24.93205
Normalized Mutual Information 0.09924 0.07788 27.41939
Hypergraph Degree Assortativity 0.02247 0.08182 72.53143
Number of Closed Triangles 115.20000 433.40000 73.41947
Number of Open Triangles 52.80000 258.00000 79.53488
Node Degree Distribution 3.52867 6.97394 49.40207
Hyperedge Size Distribution 0.30675 0.56283 45.49941
Hyper Portrait Divergence 0.11577 0.19240 39.82635
Hyper Net Simile 0.89838 1.63195 44.95066
Average Improvement 49.53423
Runtime analysis1 table
View result table
Model Training Time Sampling Time
Janus--VAE 1h 34m 7s
Janus-LDM 10h 04m -
Janus 11h 38m 42s
JanusNC--VAE 1h 18m 25s
JanusNC-LDM 9h 25m -
JanusNC 10h 43m 9m 33s

BibTeX

If you use this work, please cite the manuscript:


@inproceedings{dipasquale2026janus,
  title     = {Hypergraph Generation with Latent Diffusion},
  author    = {Di Pasquale, Valerio and Antelmi, Alessia and Polato, Mirko and Spagnuolo, Carmine},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2026},
  volume={to appear},
  doi={to appear},
  url={to appear}
}
Code repository ↗