Do Higher-Order Models Win for Higher-Order Reasons? Rethinking Performance Gains in Hypergraph Learning

Fanchen Bu; Fan Li; Geon Lee; Sunwoo Kim; Xiaoyang Wang; Renaud Lambiotte; Kijung Shin

cs.LGOct 6, 2026

Do Higher-Order Models Win for Higher-Order Reasons? Rethinking Performance Gains in Hypergraph Learning

Authors: Fanchen Bu, Fan Li, Geon Lee, Sunwoo Kim, Xiaoyang Wang, Renaud Lambiotte, Kijung Shin

Organizations: KAIST · UNSW Sydney · University of Oxford

Abstract

Higher-order models (e.g., hypergraph neural networks) often outperform lower-order baselines on hypergraph learning benchmarks, and their advantages are commonly attributed to their ability to exploit higher-order information. However, better performance alone does not establish this explanation. We therefore ask: Do higher-order models win for higher-order reasons? To investigate this question, we introduce a controlled performance-attribution framework that perturbs higher-order information while preserving the lower-order, i.e., pairwise, information. Across 25 commonly used hypergraph learning benchmarks spanning three tasks, we frequently observe an intriguing pattern: higher-order models originally outperform lower-order baselines, yet retain most of their advantage after perturbation. This suggests that much of the observed advantage remains achievable without the higher-order information. We then investigate potential lower-order explanations for these remaining gaps. We find that simple additions to a lower-order baseline, e.g., richer pairwise weighting, more steps of pairwise feature propagation, and normalization, reduce the remaining performance gaps, supporting lower-order explanations for part of the observed advantage. Our analysis calls for the hypergraph learning community to rethink performance attribution by distinguishing performance gains from their explanations, adopt stronger lower-order baselines, and use suitable benchmarks that better test the value of higher-order information.

Figures & tables

Figure 1: Large performance advantages of higher-order models can survive perturbation of higher-order information. Synthetic benchmarks show (a) a vanishing advantage of the higher-order model when higher-order information is essential and (b) equally good performance by higher- and lower-order models throughout the intervention when higher-order information is redundant. In contrast, (c–d) two real-world benchmarks exhibit a distinct pattern: higher-order models retain most of their advantages despite the perturbation. See Section 4 for details of the perturbation and Section 5 for the synthetic benchmarks and real-world results.

Figure 2: Most performance advantages of higher-order models largely persist even after the perturbation of higher-order information. Original versus fully perturbed performance gaps between higher- and lower-order models across 25 benchmarks. Of the 18 with a material original gap of at least one point, 14 retain at least 80% after perturbation. For benchmark markers, filled and half-filled symbols indicate that at least 80% or less than 80% of a material original gap remains, respectively; hollow symbols indicate an original gap below one percentage point. Stars denote synthetic cases with essential or redundant higher-order information (HOI).

Figure 3: Lower-order mechanisms such as stronger pairwise processing and matched model choices can substantially narrow the performance advantages that higher-order models retain after perturbation. Representative case studies examine the effect of (a) incorporating pair multiplicities and structural self weights, then propagating input features through pairwise connections; (b) increasing the number of feature-propagation steps and normalizing hidden features with LayerNorm; and (c) matching HGNN by removing feature-dependent messages from nodes absent from the training structure and using ELU between learned layers.

Figure 4: Simple additions to a lower-order baseline reduce its performance gap to higher-order models on fully perturbed hypergraphs, especially in node classification. For each benchmark with a largely retained gap, four bars show the gap to the higher-order reference evaluated on the fully perturbed view, after adding O, D, L, or O+D+L to CEGCN. O incorporates pair multiplicities and structural self weights; D combines the original input features with features propagated through pairwise connections; L applies LayerNorm to hidden features. Value labels report the O+D+L ratios. Each gap is divided by the gap over unmodified CEGCN:

1

(the dashed line) means no change in that gap, and

0

means matching the higher-order reference. Each task mean averages its benchmark ratios equally.

Figure 5: Cold-node handling has benchmark-specific effects. CEGCN and CEGCN+O+D+L are compared on four hyperedge-prediction benchmarks with largely retained gaps. For nodes absent from the training structure, the identity policy retains transformed features; the zero-message policy removes feature-dependent messages. All gaps use the same higher-order reference on the fully perturbed view and are divided by its gap over CEGCN using the identity policy (

1

means this gap is unchanged;

0

means matching the reference). The task mean averages the four benchmark ratios equally.

Appendix figures & tables66 assets

Supplementary material from the paper’s appendix.

Appendix

Task	Benchmark	Higher-order model	Lower-order model
EP	Actor	TF-HNN	MLP
EP	Cora-CA	UniGCNII	CEGAT
EP	DBLP-CA	PhenomNN	CEGAT
EP	Cora	PhenomNN	CEGAT
EP	Pubmed	EHNN	CEGCN
EP	Pokec	HGNN	CEGCN

Appendix

Table 1: Selected model pairs for the 25 primary benchmark comparisons. NC: node classification; EP: hyperedge prediction; HC: hypergraph classification. The Yelp pair includes the resource exception described in the results.

Task	Benchmark	$L$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$
EP	Actor	2	128	200	0.8	0.01	–	–
EP	Cora-CA	2	128	100	0.8	0.001	–	–
EP	DBLP-CA	3	128	100	0.8	0.001	–	–
EP	Cora	3	128	100	0.8	0.001	–	–
EP	Pubmed	3	128	100	0.8	0.001	–	–
EP	Pokec	2	256	100	0	0.01	–	–

Appendix

Table 2: DHG-Bench YAML settings for MLP.

Task	Benchmark	$E$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$
EP	Actor	128	64	100	0.5	0.05	false	–	–
EP	Cora-CA	128	128	100	0.5	0.01	false	–	–
EP	DBLP-CA	128	128	100	0.5	0.01	false	–	–
EP	Cora	128	256	100	0	0.001	false	–	–
EP	Pubmed	128	128	100	0.5	0.01	false	–	–
EP	Pokec	128	64	100	0.5	0.05	false	–	–

Appendix

Table 3: DHG-Bench YAML settings for CEGCN.

Task	Benchmark	$E$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$
EP	Actor	128	64	100	0.8	0.02	4	false	false	–	–
EP	Cora-CA	128	64	100	0.8	0.02	4	false	false	–	–
EP	DBLP-CA	128	64	100	0.8	0.02	4	false	false	–	–
EP	Cora	128	64	100	0.8	0.001	4	false	false	–	–
EP	Pubmed	128	64	100	0.8	0.02	4	false	false	–	–
EP	Pokec	128	64	50	0.8	0.01	4	false	false	–	–

Appendix

Table 4: DHG-Bench YAML settings for CEGAT.

Task	Benchmark	$L$	$E$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$
EP	Actor	1	128	64	400	0.5	0.01	–	–
EP	Cora-CA	2	128	128	100	0.5	0.001	–	–
EP	DBLP-CA	2	128	128	100	0.5	0.01	–	–
EP	Cora	2	128	128	100	0.5	0.001	–	–
EP	Pubmed	2	128	128	100	0.5	0.01	–	–
EP	Pokec	2	128	128	200	0	0.01	–	–

Appendix

Table 5: DHG-Bench YAML settings for HGNN.

Task	Benchmark	$L$	$E$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$
EP	Actor	1	128	64	400	0.5	0.1	true	true	–	–	–
EP	Cora-CA	2	128	128	200	0.8	0.001	true	true	false	–	–
EP	DBLP-CA	2	128	128	200	0.8	0.001	true	true	–	–	–
EP	Cora	2	128	64	150	0.5	0.001	true	true	–	–	–
EP	Pubmed	2	128	256	200	0.5	0.01	true	true	–	–	–
EP	Pokec	2	128	64	200	0.5	0.001	true	true	–	–	–

Appendix

Table 6: DHG-Bench YAML settings for HyperGCN.

Task	Benchmark	$L$	$E$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$
EP	Actor	1	128	64	400	0.5	0.01	–	–
EP	Cora-CA	2	128	128	100	0.5	0.001	–	–
EP	DBLP-CA	2	128	128	50	0.5	0.01	–	–
EP	Cora	2	128	128	100	0.5	0.001	–	–
EP	Pubmed	2	128	128	100	0.5	0.01	–	–
EP	Pokec	2	128	128	200	0	0.005	–	–

Appendix

Table 7: DHG-Bench YAML settings for HCHA.

Task	Benchmark	$L$	$E$	$H$	$T$	$p$	$\eta$	$\lambda$	$q_{1}$	$q_{2}$
EP	Actor	2	128	128	100	0.1	0.01	0.0001	–	–
EP	Cora-CA	2	128	64	100	0.1	0.01	0.01	–	–
EP	DBLP-CA	2	128	64	100	0.1	0.01	0.01	–	–
EP	Cora	2	128	256	100	0.1	0.01	0.01	–	–
EP	Pubmed	2	128	256	100	0.5	0.01	0.0001	–	–
EP	Pokec	4	128	256	200	0.1	0.001	0.0001	–	–

Appendix

Table 8: DHG-Bench YAML settings for LEGCN.

Task	Benchmark	$E$	$H$	$T$	$p$	$\eta$	$\lambda$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$	$q_{6}$	$q_{7}$
EP	Actor	128	256	300	0.5	0.001	1e-05	1	0.001	0.8	sum	–	–	–
EP	Cora-CA	128	128	200	0.5	0.001	1e-05	1	0.001	0.3	sum	–	–	–
EP	DBLP-CA	128	128	500	0.5	0.001	1e-05	1	0.001	0.3	sum	–	–	–
EP	Cora	128	128	50	0.5	0.001	1e-05	2	1e-05	0.5	sum	–	–	–
EP	Pubmed	128	512	300	0.2	0.001	1e-05	1	0.001	0.2	sum	–	–	–
EP	Pokec	128	256	300	0.5	0.01	1e-05	1	0.001	0.6	sum	–	–	–

Appendix

Table 9: DHG-Bench YAML settings for HyperND.

Task	Benchmark	$E$	$H$	$T$	$p$	$\eta$	$\lambda$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$	$q_{6}$
EP	Actor	128	128	200	0.5	0.001	1e-05	0	4	2	128	–	–
EP	Cora-CA	128	64	50	0.5	0.001	1e-05	0.1	16	1	128	–	–
EP	DBLP-CA	128	64	200	0.5	0.001	1e-05	0.1	16	1	128	–	–
EP	Cora	128	64	50	0.5	0.001	1e-05	0.5	16	1	128	–	–
EP	Pubmed	128	64	300	0.5	0.001	1e-05	0.2	16	1	128	–	–
EP	Pokec	128	64	200	0.5	0.01	1e-05	0	8	2	128	–	–

Appendix

Table 10: DHG-Bench YAML settings for PhenomNN.

Task	Benchmark	$L$	$E$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$
EP	Actor	1	128	256	200	0.5	0.001	avg	sym_degree_norm	4	–	–
EP	Cora-CA	4	128	256	15	0.5	0.001	avg	sym_degree_norm	4	–	–
EP	DBLP-CA	4	128	256	20	0.5	0.001	avg	sym_degree_norm	4	–	–
EP	Cora	3	128	512	10	0.5	0.001	avg	sym_degree_norm	4	–	–
EP	Pubmed	3	128	256	80	0.5	0.001	avg	sym_degree_norm	4	–	–
EP	Pokec	1	128	256	200	0.5	0.001	avg	sym_degree_norm	4	–	–

Appendix

Table 11: DHG-Bench YAML settings for SheafHyperGNN.

Task	Benchmark	$L$	$E$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$
EP	Actor	2	128	128	500	0.5	0.01	0.1	false	–	–
EP	Cora-CA	2	128	128	200	0.5	0.001	1	false	–	–
EP	DBLP-CA	2	128	128	100	0.5	0.001	0.1	0.1	–	–
EP	Cora	2	128	256	100	0.5	0.001	0.01	false	–	–
EP	Pubmed	2	128	128	100	0.5	0.01	0.1	false	–	–
EP	Pokec	2	128	128	500	0.1	0.001	0	false	–	–

Appendix

Table 12: DHG-Bench YAML settings for HJRL.

Task	Benchmark	$L$	$E$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$
EP	Actor	1	128	256	100	0.5	0.01	256	1	1	–	–
EP	Cora-CA	1	128	256	100	0.5	0.01	256	1	1	–	–
EP	DBLP-CA	1	128	256	100	0.5	0.01	256	1	1	–	–
EP	Cora	1	128	256	100	0.5	0.01	256	2	1	–	–
EP	Pubmed	1	128	256	100	0.5	0.01	256	1	1	–	–
EP	Pokec	1	128	256	100	0.5	0.01	256	2	2	–	–

Appendix

Table 13: DHG-Bench YAML settings for DPHGNN.

Task	Benchmark	$L$	$E$	$H$	$M$	$T$	$p$	$\eta$	$\lambda$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$
EP	Actor	2	128	512	3	100	0.1	0.001	0	1	–	–	–
EP	Cora-CA	2	128	1024	3	60	0.7	0.001	0	0.3	–	–	–
EP	DBLP-CA	2	128	1024	3	50	0.8	0.001	0	0.3	–	–	–
EP	Cora	2	128	256	3	30	0.7	0.001	0	0.1	–	–	–
EP	Pubmed	2	128	512	3	200	0.7	0.001	0	0.05	–	–	–
EP	Pokec	2	128	1024	3	150	0.5	0.01	0.0001	0.9	–	–	–

Appendix

Table 14: DHG-Bench YAML settings for TF-HNN.

Task	Benchmark	$L$	$E$	$H$	$T$	$p$	$\eta$	$\lambda$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$
EP	Actor	1	256	128	300	0.5	0.001	0.0001	-1.5	-0.5	true	–	–
EP	Cora-CA	1	256	256	100	0.5	0.001	1e-05	-1.5	-0.5	true	–	–
EP	DBLP-CA	1	256	256	100	0.5	0.001	0.0001	-1.5	-0.5	true	–	–
EP	Cora	1	256	128	200	0.5	0.001	0.0001	-1.5	-0.5	true	–	–
EP	Pubmed	1	256	512	50	0.5	0.001	0.0001	-1.5	-0.5	false	–	–
EP	Pokec	1	256	128	200	0.5	0.01	0.0001	-1.5	-0.5	true	–	–

Appendix

Table 15: DHG-Bench YAML settings for HNHN.

Task	Benchmark	$L$	$E$	$H$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$	$q_{6}$	$q_{7}$
EP	Actor	1	–	256	200	0.5	0.01	0.5	prelu	1	0.5	mean	–	–
EP	Cora-CA	2	–	256	50	0.8	0.001	0.5	prelu	0.3	0.8	mean	–	–
EP	DBLP-CA	2	–	128	100	0.5	0.001	0.3	prelu	0.2	0.3	mean	–	–
EP	Cora	2	–	256	50	0.8	0.001	0.5	prelu	0.3	0.8	mean	–	–
EP	Pubmed	2	–	128	100	0	0.001	0.3	prelu	0.5	0.5	mean	–	–
EP	Pokec	1	–	128	100	0	0.001	0.5	prelu	0.5	0.5	mean	–	–

Appendix

Table 16: DHG-Bench YAML settings for UniGCNII.

Task	Benchmark	$L$	$E$	$H$	$M$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$
EP	Actor	2	128	128	2	200	0.5	0.001	true	128	2	–	–
EP	Cora-CA	1	128	256	2	300	0.5	0.001	true	128	2	–	–
EP	DBLP-CA	2	128	128	2	50	0.5	0.001	true	128	2	–	–
EP	Cora	2	128	256	2	100	0.5	0.001	true	64	2	–	–
EP	Pubmed	1	128	256	2	300	0.5	0.001	true	64	2	–	–
EP	Pokec	2	128	64	2	200	0.5	0.001	true	128	2	–	–

Appendix

Table 17: DHG-Bench YAML settings for AllSetTransformer.

Task	Benchmark	$L$	$E$	$H$	$M$	$T$	$p$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$	$q_{6}$	$q_{7}$
EP	Actor	1	128	512	0	180	0.5	0.001	1	1	256	2	0.2	–	–
EP	Cora-CA	1	128	512	0	500	0.5	0.001	1	1	256	1	0.1	–	–
EP	DBLP-CA	1	128	512	1	40	0.5	0.001	1	1	256	2	0.7	–	–
EP	Cora	1	128	512	0	500	0.5	0.001	1	1	256	1	0.5	–	–
EP	Pubmed	1	128	512	0	100	0.5	0.001	1	1	256	1	0.8	–	–
EP	Pokec	1	128	512	0	140	0.5	0.001	1	1	256	1	0.2	–	–

Appendix

Table 18: DHG-Bench YAML settings for ED-HNN.

Task	Benchmark	$L$	$E$	$T$	$p$	$\eta$	$\lambda$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$	$q_{5}$
EP	Actor	2	128	100	0.5	0.005	0.001	128	false	max	–	–
EP	Cora-CA	2	128	100	0.5	0.005	0.001	256	false	maxmin	–	–
EP	DBLP-CA	2	128	100	0.5	0.0005	0.01	128	false	maxmin	–	–
EP	Cora	2	128	100	0.5	0.005	0.001	128	true	maxmin	–	–
EP	Pubmed	2	128	100	0.5	0.01	0.001	128	true	maxmin	–	–
EP	Pokec	2	128	100	0.5	0.01	0.01	128	false	maxmin	–	–

Appendix

Table 19: DHG-Bench YAML settings for HyperGT.

Task	Benchmark	$H$	$T$	$p$	$\lambda$	$q_{1}$
EP	Actor	128	500	0.2	1e-05	4
EP	Cora-CA	128	100	0	0.001	8
EP	DBLP-CA	128	200	0	1e-05	4
EP	Cora	128	80	0.2	1e-05	8
EP	Pubmed	128	200	0	1e-05	8
EP	Pokec	128	180	0	0	4

Appendix

Table 20: DHG-Bench YAML settings for EHNN.

Task	Benchmark	$H$	$T$	$\eta$	$q_{1}$	$q_{2}$	$q_{3}$	$q_{4}$
EP	Actor	256	50	0.01	max	10	256	[5]
EP	Cora-CA	256	60	0.01	max	10	256	[5]
EP	DBLP-CA	256	40	0.01	max	10	128	[5]
EP	Cora	128	80	0.01	max	10	128	[5]
EP	Pubmed	256	60	0.01	max	10	256	[5]
EP	Pokec	256	80	0.01	max	1	256	[5]

Appendix

Table 21: DHG-Bench YAML settings for T-HyperGNN.

Task	Dataset	Fixed pair	$\Delta^{(0)}$	$\Delta^{(0.5)}$	$\Delta^{(1)}$
Node classification	ModelNet40	EDHNN–MLP	+2.00	+1.85	+1.87
Node classification	NTU2012	LEGCN–MLP	+2.45	+1.68	+1.39
Node classification	Actor	EHNN–MLP	+0.42	+0.07	-0.27
Node classification	Cora-CA	TFHNN–CEGCN	+6.50	+6.15	+6.40
Node classification	DBLP-CA	PhenomNN–CEGCN	+2.12	+2.13	+2.05
Node classification	Cora	TFHNN–CEGCN	+4.38	+4.38	+4.52

Appendix

Table 22: Complete selected-pair gaps.

\Delta

is the higher-order model’s test performance minus that of the fixed lower-order comparator, in percentage points. The metric is accuracy for node classification and hypergraph classification and AP for hyperedge prediction.

Task	Dataset	Original gap	Endpoint gap	Retained (%)
Node classification	Actor †	0.418	-0.271	-64.7
Node classification	Cora	4.376	4.523	103.4
Node classification	Cora-CA	6.499	6.401	98.5
Node classification	DBLP-CA	2.118	2.053	97.0
Node classification	Pokec †	0.089	-0.551	-620.0
Node classification	Pubmed	1.487	1.467	98.6

Appendix

Table 23: Gap retention for every benchmark. Gaps are accuracy points (node and hypergraph classification) or AP points (hyperedge prediction). Retention is

100\Delta^{(1)}/\Delta^{(0)}

, displayed to one decimal place.

\dagger

marks a positive original gap below one point. N/A denotes a nonpositive original gap.

Figure 6: Original and fully perturbed performance gaps using AUROC for hyperedge prediction and Macro-F1 for hypergraph classification. Checkpoints and model pairs are selected using the corresponding secondary validation metric. Blue circles indicate original gaps of at least one percentage point that retain at least 80% after perturbation; orange squares denote the other comparisons. The dotted diagonal denotes unchanged gaps and the dashed line 80% retention.

Task	Dataset	Higher-order	Lower-order	$\Delta^{(0)}$	$\Delta^{(1)}$	Retention
Hyperedge prediction	Actor	TF-HNN	MLP	6.46	6.46	100.0%
Hyperedge prediction	Cora	PhenomNN	CEGAT	$-1.04$	$-1.11$	–
Hyperedge prediction	Cora-CA	UniGCNII	CEGAT	3.94	3.04	77.2%
Hyperedge prediction	DBLP-CA	PhenomNN	CEGAT	0.56	0.69	–
Hyperedge prediction	Pokec	HGNN	CEGCN	24.50	25.17	102.7%
Hyperedge prediction	Pubmed	EHNN	CEGCN	$-2.09$	$-4.52$	–

Appendix

Table 24: Secondary-metric gaps after selecting checkpoints and model pairs using the corresponding original-view validation metric: AUROC for hyperedge prediction and Macro-F1 for hypergraph classification. Model identities and hyperparameters remain fixed across views. Gaps are measured in percentage points. Retention is reported only when

\Delta^{(0)}\geq 1

point; “–” denotes an ineligible comparison.

Benchmark	Model	$P^{(0)}$	$P^{(1)}$	Max	Frozen
Hyperedge prediction
Actor	MLP	72.46	72.46	0.000	$0$
Cora-CA	CEGAT	63.31	64.09	1.785	$2.38\times 10^{-7}$
DBLP-CA	CEGAT	73.94	73.92	1.983	$6.20\times 10^{-6}$
Cora	CEGAT	65.75	65.35	0.636	$3.28\times 10^{-7}$
Pubmed	CEGCN	71.11	71.43	0.961	$3.58\times 10^{-7}$

Appendix

Table 25: Independent retraining of the selected lower-order baseline on original and fully perturbed views. Scores are three-seed means (percent). Max is the largest absolute seed-level score difference. Frozen compares output logits or query scores across views using the same checkpoint, reporting the largest absolute difference across both source views and three seeds.

Benchmark	Model	$P^{(0)}$	$P^{(1)}$	Max
Hyperedge prediction
Actor	MLP	76.58	76.58	0.000
Cora-CA	CEGAT	66.13	66.62	1.720
DBLP-CA	CEGAT	75.38	76.12	1.690
Cora	CEGAT	69.24	68.47	1.090
Pubmed	CEGCN	74.59	74.36	0.361

Appendix

Table 26: Secondary scores from the independent baseline audit: AUROC for hyperedge prediction and Macro-F1 for hypergraph classification.

Figure 7: Original and fully perturbed performance for every higher-order model on the first seven node-classification benchmarks. Entries are test-accuracy percentages. Color is clipped at $\pm 10$ points while the annotations retain the exact values.

Figure 8: Original and fully perturbed performance for every higher-order model on the remaining six node-classification benchmarks. Styling follows Fig. 7 .

Figure 9: Performance at five perturbation levels for the three higher-order models with the best original-view validation performance on the first seven node-classification benchmarks. Rank colors, markers, and line styles are shared across panels; each panel identifies the corresponding model names.

Figure 10: Performance at five perturbation levels for the three higher-order models with the best original-view validation performance on the remaining six node-classification benchmarks. Styling and aggregation follow Fig. 9 .

Figure 11: Original and fully perturbed AP for every higher-order model supporting hyperedge prediction. Each cell gives original $\rightarrow$ fully perturbed test performance ( $P^{(0)}\!\rightarrow P^{(1)}$ ), in percent. The background encodes their difference, black borders mark decreases of at least one point, and gray cells are unavailable.

Figure 12: Performance at five perturbation levels for the three higher-order models with the best original-view validation performance on all six hyperedge-prediction benchmarks. The black dotted line is the same selected lower-order model used in Fig. 2 .

Figure 13: Original and fully perturbed performance for all higher-order models using AUROC. Each cell shows original $\rightarrow$ fully perturbed performance (percent), with checkpoints selected by validation AUROC. Color denotes the paired decrease in points. Outlines mark decreases of at least one point. Gray cells indicate unavailable comparisons; a dagger marks fewer than three paired seeds.

Figure 14: Five-level AUROC trajectories. The top three HNNs and the lower-order comparator are selected by original-view validation AUROC; checkpoints at every level are also selected by validation AUROC. HNN error bars show sample SD across available seed means (three seeds planned), after averaging the three perturbation trajectories within each seed at intermediate levels. Crosses mark unavailable cells.

Figure 15: Original and fully perturbed accuracy for every higher-order model supporting hypergraph classification. Each cell gives original $\rightarrow$ fully perturbed test performance ( $P^{(0)}\!\rightarrow P^{(1)}$ ), in percent. The background encodes their difference, black borders mark decreases of at least one point, and gray cells are unavailable.

Figure 16: Performance at five perturbation levels for the three higher-order models with the best original-view validation performance on all six hypergraph-classification benchmarks. The black dotted line is the same selected lower-order model used in Fig. 2 .

Figure 17: Original and fully perturbed performance for all higher-order models using Macro-F1. Original $\rightarrow$ fully perturbed performance (percent), with checkpoints selected by validation Macro-F1. Color denotes the paired decrease. Outlines mark decreases of at least one point. Gray cells indicate unavailable comparisons; a dagger marks fewer than three paired seeds.

Figure 18: Five-level Macro-F1 trajectories. The top three HNNs and the lower-order comparator are selected by original-view validation Macro-F1; checkpoints at every level are also selected by validation Macro-F1. HNN error bars show sample SD across available seed means (three seeds planned), after averaging the three perturbation trajectories within each seed at intermediate levels. Crosses mark unavailable cells.

Extra steps $k$	Walmart: $W^{k}$	Trivago: $P^{k}$	Trivago: $W^{k}$
0	$74.56\pm 0.19$	$47.75\pm 0.52$	$47.75\pm 0.52$
1	$76.32\pm 0.23$	$52.90\pm 0.23$	$50.05\pm 0.50$
2	$75.68\pm 0.37$	$55.72\pm 0.28$	$51.98\pm 0.07$
4	$72.20\pm 0.58$	$59.84\pm 0.35$	$54.72\pm 0.28$
8	$64.43\pm 0.47$	$64.95\pm 0.22$	$58.49\pm 0.37$
16	$49.78\pm 0.34$	$70.45\pm 0.15$	$63.49\pm 0.30$

Appendix

Table 27: Input-diffusion depth sensitivity for the Walmart and Trivago controls. Each entry reports test accuracy as mean

\pm

sample SD. Diffusion is applied to the input features before the fixed CEGCN classifier;

k=0

denotes no extra diffusion.

W

is the operator incorporating pair multiplicities and structural self weights, while

P

is the binary CEGCN operator.

Classifier change	$k=64$	$k=256$
Matched baseline	$74.41\pm 0.15$	$78.32\pm 0.65$
+ LayerNorm	$87.96\pm 0.96$	$90.92\pm 0.75$
Weighted propagation throughout	$73.74\pm 0.19$	$77.37\pm 0.91$
Remove classifier propagation	$72.94\pm 0.43$	$75.83\pm 0.91$

Appendix

Table 28: Classifier controls on Trivago with the same propagated input features

W^{k}X

, where

W

incorporates pair multiplicities and structural self weights. Entries report test accuracy as mean

\pm

sample SD. The matched baseline retains binary propagation in the two learned CEGCN layers; each subsequent row changes one classifier component while keeping the remaining settings fixed.

Encoder change	Test AP (%)
Matched cold-node policy	$78.22\pm 1.73$
+ ELU	$80.14\pm 2.11$
+ LayerNorm	$79.70\pm 0.48$
+ count-mean propagation	$78.48\pm 1.73$

Appendix

Table 29: Complementary encoder controls on Pokec after matching the higher-order reference’s cold-node policy. Entries report test AP as mean

\pm

sample SD. Each subsequent row adds one encoder change to the CEGCN baseline using the same zero-message policy as HGNN.

Model	Computation on the fully perturbed endpoint	Weighting and architectural details
HGNN	Learned feature maps with $Q_{\mathrm{mean}}$ propagation and ELU between layers.	Averaging through each pair induces both neighbor and self messages; repeated pairs and singleton incidences enter aggregation and degree normalization.
HCHA	The released implementation uses the same convolution class as HGNN, with $Q_{\mathrm{sym}}$ propagation.	Symmetric degree normalization replaces row normalization; the attention branch is not enabled by the model constructor.
HyperGCN	Graph convolution on the graph constructed from endpoint hyperedges by the extremal-node/mediator routine.	Edge accumulation, normalization, explicit self loops, singleton handling, and feature-dependent tie handling follow the released implementation.
TFHNN	Fixed input diffusion using $\mathcal{N}(K+I)$ and the configured restart, followed by an MLP decoder.	Pair counts, singleton multiplicities, and degree-dependent self weights enter diffusion; restart weight $\alpha=1$ gives a feature-only configuration, as on Actor hyperedge prediction.
UniGCNII	Mean aggregation through endpoint hyperedges, optional feature normalization, restart to the initial representation, and learned residual maps.	Uses $Q_{\mathrm{mean}}$ ; the restart path can preserve features even when the structural aggregate is zero.
PhenomNN	Iterative propagation with two normalized matrices determined by pair and singleton incidences, feature injection, and ReLU, between an encoder and decoder.	Both $C+\operatorname{diag}(q)+I$ and $K+I$ enter; setting the propagation depth to zero retains feature processing.

Appendix

Table 30: Computations of the 17 HNNs on fully perturbed endpoints. Pair and singleton multiplicities are retained in the endpoint inputs. The table identifies propagation, normalization, and architectural choices; singleton handling and additional self loops follow each implementation.

Dataset	CEGCN	+O	+D	+L	+O+D+L
ModelNet40	$92.75\pm 0.56$	$97.85\pm 0.42$	$92.63\pm 0.66$	$92.61\pm 0.64$	$97.45\pm 0.47$
NTU2012	$84.76\pm 0.64$	$88.87\pm 0.72$	$84.63\pm 0.75$	$84.10\pm 1.05$	$88.87\pm 1.30$
Actor	$67.86\pm 0.42$	$82.26\pm 0.53$	$67.77\pm 0.47$	$68.23\pm 0.31$	$82.44\pm 0.36$
Cora-CA	$79.57\pm 2.79$	$84.59\pm 0.45$	$79.67\pm 3.13$	$79.76\pm 2.74$	$84.64\pm 0.15$
DBLP-CA	$89.83\pm 0.15$	$91.19\pm 0.35$	$89.81\pm 0.07$	$90.24\pm 0.17$	$91.73\pm 0.19$
Cora	$77.73\pm 2.23$	$78.52\pm 1.73$	$78.07\pm 2.00$	$76.55\pm 3.32$	$78.96\pm 3.63$

Appendix

Table 31: Node classification. Mean

\pm

sample SD. The metric is accuracy.

Dataset	+O	+D	+L	+O+D+L
ModelNet40	$+5.10\pm 0.34$	$-0.12\pm 0.11$	$-0.14\pm 0.21$	$+4.70\pm 0.27$
NTU2012	$+4.11\pm 0.61$	$-0.13\pm 0.11$	$-0.66\pm 0.50$	$+4.11\pm 0.98$
Actor	$+14.40\pm 0.40$	$-0.10\pm 0.05$	$+0.37\pm 0.23$	$+14.58\pm 0.37$
Cora-CA	$+5.02\pm 2.60$	$+0.10\pm 0.70$	$+0.20\pm 0.37$	$+5.07\pm 2.74$
DBLP-CA	$+1.37\pm 0.22$	$-0.01\pm 0.15$	$+0.41\pm 0.03$	$+1.90\pm 0.21$
Cora	$+0.79\pm 1.34$	$+0.34\pm 0.61$	$-1.18\pm 1.29$	$+1.23\pm 1.51$

Appendix

Table 32: Node classification. Changes relative to the official CEGCN, mean

\pm

sample SD.

Dataset	CEGCN	+O 1	+O 1 +O 2	$\delta_{1}$	$\delta_{2}$
ModelNet40	$92.75\pm 0.56$	$92.93\pm 0.86$	$97.85\pm 0.42$	$0.18\pm 0.41$	$4.92\pm 0.47$
NTU2012	$84.76\pm 0.64$	$85.69\pm 0.72$	$88.87\pm 0.72$	$0.93\pm 0.61$	$3.18\pm 0.00$
Actor	$67.86\pm 0.42$	$67.59\pm 0.51$	$82.26\pm 0.53$	$-0.27\pm 0.22$	$14.67\pm 0.21$
Cora-CA	$79.57\pm 2.79$	$80.01\pm 2.37$	$84.59\pm 0.45$	$0.44\pm 0.53$	$4.58\pm 2.13$
DBLP-CA	$89.83\pm 0.15$	$89.91\pm 0.15$	$91.19\pm 0.35$	$0.09\pm 0.05$	$1.28\pm 0.25$
Cora	$77.73\pm 2.23$	$78.17\pm 2.45$	$78.52\pm 1.73$	$0.44\pm 0.29$	$0.34\pm 1.63$

Appendix

Table 33: Node classification: successive changes to the propagation operator (accuracy). Original-view scores (percent) and paired stepwise changes (percentage points) are mean

\pm

sample SD. O 1 adds pair multiplicities; O 2 changes structural self weights and normalization after pair multiplicities have been added by O 1 . D and L are disabled.

Dataset	CEGCN	+O	+D	+L	+O+D+L
IMDB-Dir-Form	$63.30\pm 1.92$	$61.70\pm 0.53$	$63.83\pm 3.72$	$61.52\pm 1.11$	$60.28\pm 2.01$
IMDB-Dir-Genre	$77.55\pm 1.11$	$76.86\pm 1.45$	$77.35\pm 1.35$	$78.24\pm 0.51$	$76.47\pm 2.35$
RHG-10	$95.00\pm 1.80$	$93.83\pm 1.89$	$95.00\pm 1.32$	$95.83\pm 0.76$	$94.67\pm 0.29$
RHG-3	$98.67\pm 0.67$	$98.00\pm 1.76$	$98.00\pm 0.67$	$98.44\pm 0.77$	$99.78\pm 0.38$
Steam-Player	$54.85\pm 2.43$	$55.18\pm 1.01$	$54.53\pm 2.67$	$53.24\pm 4.04$	$53.72\pm 2.67$
Twitter-Friend	$58.02\pm 1.32$	$56.23\pm 3.09$	$51.40\pm 1.59$	$55.73\pm 3.05$	$56.74\pm 1.17$

Appendix

Table 34: Hypergraph classification. Mean

\pm

sample SD. The metric is accuracy.

Dataset	+O	+D	+L	+O+D+L
IMDB-Dir-Form	$-1.60\pm 1.60$	$+0.53\pm 2.76$	$-1.77\pm 1.63$	$-3.01\pm 3.74$
IMDB-Dir-Genre	$-0.69\pm 0.34$	$-0.20\pm 1.03$	$+0.69\pm 0.90$	$-1.08\pm 1.62$
RHG-10	$-1.17\pm 1.15$	$+0.00\pm 1.50$	$+0.83\pm 2.25$	$-0.33\pm 1.61$
RHG-3	$-0.67\pm 2.00$	$-0.67\pm 0.67$	$-0.22\pm 0.38$	$+1.11\pm 0.38$
Steam-Player	$+0.32\pm 1.48$	$-0.32\pm 0.28$	$-1.62\pm 2.84$	$-1.13\pm 0.28$
Twitter-Friend	$-1.78\pm 3.92$	$-6.62\pm 2.68$	$-2.29\pm 4.25$	$-1.27\pm 2.45$

Appendix

Table 35: Hypergraph classification. Changes relative to the official CEGCN, mean

\pm

sample SD.

Dataset	CEGCN	+O 1	+O 1 +O 2	$\delta_{1}$	$\delta_{2}$
IMDB-Dir-Form	$63.30\pm 1.92$	$60.28\pm 2.68$	$61.70\pm 0.53$	$-3.01\pm 3.02$	$1.42\pm 3.02$
IMDB-Dir-Genre	$77.55\pm 1.11$	$77.94\pm 0.51$	$76.86\pm 1.45$	$0.39\pm 1.03$	$-1.08\pm 1.33$
RHG-10	$95.00\pm 1.80$	$94.67\pm 0.58$	$93.83\pm 1.89$	$-0.33\pm 2.02$	$-0.83\pm 1.76$
RHG-3	$98.67\pm 0.67$	$97.78\pm 1.39$	$98.00\pm 1.76$	$-0.89\pm 0.77$	$0.22\pm 2.69$
Steam-Player	$54.85\pm 2.43$	$54.05\pm 2.93$	$55.18\pm 1.01$	$-0.81\pm 0.56$	$1.13\pm 2.02$
Twitter-Friend	$58.02\pm 1.32$	$55.22\pm 5.93$	$56.23\pm 3.09$	$-2.80\pm 7.21$	$1.02\pm 5.93$

Appendix

Table 36: Hypergraph classification: successive changes to the propagation operator (accuracy). Original-view scores (percent) and paired stepwise changes (percentage points) are mean

\pm

sample SD. O 1 adds pair multiplicities; O 2 changes structural self weights and normalization after pair multiplicities have been added by O 1 . D and L are disabled.

Dataset	CEGCN	+O	+D	+L	+O+D+L
Actor	$72.11\pm 0.57$	$74.91\pm 1.75$	$71.39\pm 1.76$	$69.39\pm 0.90$	$75.39\pm 0.79$
Cora-CA	$59.57\pm 1.67$	$58.87\pm 1.25$	$59.95\pm 2.46$	$57.75\pm 1.30$	$59.55\pm 2.00$
DBLP-CA	$63.80\pm 1.34$	$73.03\pm 2.23$	$63.34\pm 2.39$	$63.16\pm 0.50$	$71.59\pm 1.00$
Cora	$59.27\pm 1.18$	$57.99\pm 1.82$	$59.02\pm 1.49$	$56.45\pm 1.35$	$57.99\pm 1.18$
Pubmed	$71.12\pm 0.70$	$71.49\pm 0.72$	$71.89\pm 0.27$	$70.42\pm 0.51$	$71.75\pm 1.68$
Pokec	$59.51\pm 3.01$	$55.17\pm 1.40$	$60.11\pm 3.73$	$59.56\pm 2.45$	$59.74\pm 0.74$

Appendix

Table 37: Hyperedge prediction using the identity policy for cold nodes. Mean

\pm

sample SD. The metric is AP.

Dataset	+O	+D	+L	+O+D+L
Actor	$+2.80\pm 2.14$	$-0.72\pm 2.01$	$-2.72\pm 0.73$	$+3.28\pm 1.29$
Cora-CA	$-0.70\pm 1.50$	$+0.38\pm 1.70$	$-1.82\pm 0.37$	$-0.02\pm 3.13$
DBLP-CA	$+9.23\pm 1.43$	$-0.46\pm 1.29$	$-0.64\pm 1.21$	$+7.79\pm 0.69$
Cora	$-1.28\pm 1.49$	$-0.25\pm 0.83$	$-2.82\pm 0.95$	$-1.28\pm 1.05$
Pubmed	$+0.37\pm 0.31$	$+0.77\pm 0.85$	$-0.70\pm 1.21$	$+0.63\pm 1.02$
Pokec	$-4.34\pm 1.76$	$+0.60\pm 1.83$	$+0.05\pm 3.49$	$+0.23\pm 3.52$

Appendix

Table 38: Hyperedge prediction using the identity policy for cold nodes: changes relative to the official CEGCN using the same policy for cold nodes, mean

\pm

sample SD.

Dataset	CEGCN	+O 1	+O 1 +O 2	$\delta_{1}$	$\delta_{2}$
Actor	$72.11\pm 0.57$	$71.08\pm 2.16$	$74.91\pm 1.75$	$-1.03\pm 1.63$	$3.83\pm 3.73$
Cora-CA	$59.57\pm 1.67$	$59.17\pm 1.32$	$58.87\pm 1.25$	$-0.40\pm 0.87$	$-0.30\pm 1.83$
DBLP-CA	$63.80\pm 1.34$	$64.18\pm 2.10$	$73.03\pm 2.23$	$0.38\pm 0.97$	$8.85\pm 0.66$
Cora	$59.27\pm 1.18$	$58.85\pm 1.01$	$57.99\pm 1.82$	$-0.42\pm 0.19$	$-0.85\pm 1.39$
Pubmed	$71.12\pm 0.70$	$71.85\pm 0.60$	$71.49\pm 0.72$	$0.73\pm 0.19$	$-0.36\pm 0.17$
Pokec	$59.51\pm 3.01$	$59.29\pm 1.15$	$55.17\pm 1.40$	$-0.22\pm 3.37$	$-4.12\pm 2.20$

Appendix

Table 39: Hyperedge prediction: successive changes to the propagation operator (AP; identity policy for cold nodes). Original-view scores (percent) and paired stepwise changes (percentage points) are mean

\pm

sample SD. O 1 adds pair multiplicities; O 2 changes structural self weights and normalization after pair multiplicities have been added by O 1 . D and L are disabled.

Dataset	CEGCN	+O	+D	+L	+O+D+L
Actor	$64.97\pm 1.39$	$67.46\pm 1.85$	$64.09\pm 0.36$	$65.14\pm 1.33$	$67.87\pm 2.50$
Cora-CA	$60.85\pm 7.06$	$56.01\pm 2.27$	$60.89\pm 5.55$	$60.77\pm 3.60$	$60.30\pm 0.51$
DBLP-CA	$65.19\pm 3.46$	$60.91\pm 0.34$	$65.73\pm 3.48$	$69.85\pm 1.42$	$71.49\pm 0.25$
Cora	$57.52\pm 0.79$	$56.24\pm 0.80$	$57.13\pm 0.97$	$57.09\pm 1.09$	$57.52\pm 0.75$
Pubmed	$69.38\pm 1.14$	$68.99\pm 1.08$	$69.84\pm 0.57$	$68.74\pm 1.23$	$69.21\pm 0.58$
Pokec	$78.37\pm 2.05$	$75.81\pm 2.99$	$77.93\pm 1.01$	$79.37\pm 0.81$	$78.34\pm 1.71$

Appendix

Table 40: Hyperedge prediction using the zero-message policy for cold nodes. Mean

\pm

sample SD. The metric is AP.

Dataset	+O	+D	+L	+O+D+L
Actor	$+2.49\pm 3.21$	$-0.87\pm 1.65$	$+0.17\pm 2.69$	$+2.90\pm 1.24$
Cora-CA	$-4.84\pm 4.94$	$+0.04\pm 1.79$	$-0.08\pm 7.11$	$-0.55\pm 6.55$
DBLP-CA	$-4.28\pm 3.61$	$+0.54\pm 1.15$	$+4.65\pm 2.16$	$+6.30\pm 3.21$
Cora	$-1.28\pm 0.31$	$-0.39\pm 0.60$	$-0.43\pm 1.18$	$-0.00\pm 1.24$
Pubmed	$-0.39\pm 0.09$	$+0.46\pm 1.06$	$-0.64\pm 0.85$	$-0.17\pm 1.12$
Pokec	$-2.56\pm 2.85$	$-0.44\pm 1.64$	$+1.00\pm 1.30$	$-0.03\pm 0.34$

Appendix

Table 41: Hyperedge prediction using the zero-message policy for cold nodes. Changes relative to the official CEGCN using the same policy for cold nodes, mean

\pm

sample SD.

Dataset	CEGCN	+O 1	+O 1 +O 2	$\delta_{1}$	$\delta_{2}$
Actor	$64.97\pm 1.39$	$63.42\pm 0.65$	$67.46\pm 1.85$	$-1.55\pm 1.02$	$4.04\pm 2.45$
Cora-CA	$60.85\pm 7.06$	$59.66\pm 5.67$	$56.01\pm 2.27$	$-1.18\pm 1.99$	$-3.65\pm 3.89$
DBLP-CA	$65.19\pm 3.46$	$61.70\pm 2.29$	$60.91\pm 0.34$	$-3.50\pm 2.92$	$-0.79\pm 2.63$
Cora	$57.52\pm 0.79$	$57.17\pm 1.01$	$56.24\pm 0.80$	$-0.35\pm 0.83$	$-0.92\pm 1.12$
Pubmed	$69.38\pm 1.14$	$69.72\pm 1.16$	$68.99\pm 1.08$	$0.34\pm 1.03$	$-0.73\pm 0.95$
Pokec	$78.37\pm 2.05$	$75.35\pm 0.62$	$75.81\pm 2.99$	$-3.01\pm 1.53$	$0.46\pm 2.52$

Appendix

Table 42: Hyperedge prediction: successive changes to the propagation operator (AP; zero-message policy for cold nodes). Original-view scores (percent) and paired stepwise changes (percentage points) are mean

\pm

sample SD. O 1 adds pair multiplicities; O 2 changes structural self weights and normalization after pair multiplicities have been added by O 1 . D and L are disabled.

Dataset	CEGCN	+O	+D	+L	+O+D+L
Actor	$-7.14\pm 1.37$	$-7.45\pm 3.03$	$-7.30\pm 1.52$	$-4.25\pm 1.25$	$-7.51\pm 2.55$
Cora-CA	$+1.28\pm 6.03$	$-2.87\pm 3.14$	$+0.94\pm 5.72$	$+3.02\pm 4.46$	$+0.75\pm 2.48$
DBLP-CA	$+1.39\pm 2.38$	$-12.12\pm 2.01$	$+2.39\pm 1.29$	$+6.69\pm 1.19$	$-0.10\pm 0.90$
Cora	$-1.75\pm 0.89$	$-1.75\pm 2.38$	$-1.89\pm 0.70$	$+0.64\pm 0.43$	$-0.47\pm 0.44$
Pubmed	$-1.74\pm 0.58$	$-2.51\pm 0.86$	$-2.05\pm 0.35$	$-1.68\pm 1.43$	$-2.54\pm 1.95$
Pokec	$+18.86\pm 3.29$	$+20.64\pm 3.95$	$+17.82\pm 2.72$	$+19.81\pm 3.12$	$+18.60\pm 2.38$

Appendix

Table 43: Hyperedge prediction: AP under the zero-message versus identity policy for cold nodes, mean

\pm

sample SD.

Task / policy	+O	+D	+L	+O+D+L
Edge-Pred: identity (AUROC)	$+1.69$ (3/6)	$-0.04$ (4/6)	$-1.58$ (1/6)	$+2.31$ (4/6)
Edge-Pred: zero (AUROC)	$-2.73$ (0/6)	$-0.55$ (2/6)	$+0.34$ (1/6)	$+1.07$ (4/6)
HG-CLS (Macro-F1)	$-2.62$ (0/6)	$-1.29$ (2/6)	$-0.58$ (3/6)	$-0.78$ (1/6)

Appendix

Table 44: Secondary-metric gains on all benchmarks. Edge-Pred denotes hyperedge prediction; HG-CLS denotes hypergraph classification. Mean paired gain over standard CEGCN (points), weighting benchmarks equally and using the same policy for cold nodes in hyperedge prediction; followed by the number of benchmarks with a positive mean gain.

Dataset	CEGCN	+O	+D	+L	+O+D+L
IMDB-Dir-Form	$57.19\pm 2.34$	$56.45\pm 3.18$	$58.09\pm 5.91$	$57.66\pm 1.48$	$56.15\pm 4.49$
IMDB-Dir-Genre	$72.65\pm 2.55$	$70.60\pm 4.08$	$72.58\pm 2.79$	$73.07\pm 2.08$	$71.28\pm 2.93$
RHG-10	$94.81\pm 1.88$	$93.85\pm 2.00$	$94.83\pm 1.32$	$95.68\pm 0.64$	$94.55\pm 0.39$
RHG-3	$98.72\pm 0.64$	$98.00\pm 1.78$	$98.10\pm 0.54$	$98.52\pm 0.70$	$99.78\pm 0.39$
Steam-Player	$54.03\pm 2.57$	$47.68\pm 1.94$	$52.81\pm 3.64$	$52.24\pm 3.03$	$51.43\pm 1.52$
Twitter-Friend	$54.44\pm 5.46$	$49.56\pm 2.25$	$47.72\pm 4.07$	$51.22\pm 1.92$	$53.98\pm 2.62$

Appendix

Table 45: Hypergraph classification (Macro-F1): original-view scores (percent), mean

\pm

sample SD.

Dataset	+O	+D	+L	+O+D+L
IMDB-Dir-Form	$-0.73\pm 0.84$	$+0.90\pm 3.66$	$+0.48\pm 0.89$	$-1.03\pm 2.94$
IMDB-Dir-Genre	$-2.05\pm 1.55$	$-0.07\pm 1.54$	$+0.41\pm 1.19$	$-1.37\pm 1.87$
RHG-10	$-0.96\pm 1.18$	$+0.02\pm 1.47$	$+0.87\pm 2.28$	$-0.26\pm 1.61$
RHG-3	$-0.72\pm 1.97$	$-0.62\pm 0.61$	$-0.20\pm 0.35$	$+1.05\pm 0.33$
Steam-Player	$-6.36\pm 2.17$	$-1.23\pm 1.75$	$-1.79\pm 1.50$	$-2.61\pm 1.84$
Twitter-Friend	$-4.89\pm 3.30$	$-6.72\pm 3.48$	$-3.23\pm 4.49$	$-0.47\pm 7.38$

Appendix

Table 46: Hypergraph classification (Macro-F1): paired gains over standard CEGCN (points), mean

\pm

sample SD.

Dataset	CEGCN	+O 1	+O 1 +O 2	$\delta_{1}$	$\delta_{2}$
IMDB-Dir-Form	$57.19\pm 2.34$	$54.61\pm 1.08$	$56.45\pm 3.18$	$-2.58\pm 2.06$	$1.85\pm 2.83$
IMDB-Dir-Genre	$72.65\pm 2.55$	$73.20\pm 1.38$	$70.60\pm 4.08$	$0.54\pm 1.59$	$-2.60\pm 2.94$
RHG-10	$94.81\pm 1.88$	$94.57\pm 0.63$	$93.85\pm 2.00$	$-0.24\pm 2.14$	$-0.73\pm 1.88$
RHG-3	$98.72\pm 0.64$	$97.82\pm 1.35$	$98.00\pm 1.78$	$-0.90\pm 0.78$	$0.18\pm 2.68$
Steam-Player	$54.03\pm 2.57$	$52.60\pm 3.32$	$47.68\pm 1.94$	$-1.44\pm 1.10$	$-4.92\pm 2.20$
Twitter-Friend	$54.44\pm 5.46$	$51.61\pm 3.51$	$49.56\pm 2.25$	$-2.83\pm 7.24$	$-2.06\pm 4.26$

Appendix

Table 47: Hypergraph classification: successive changes to the propagation operator (Macro-F1). Original-view scores (percent) and paired stepwise changes (percentage points) are mean

\pm

sample SD. O 1 adds pair multiplicities; O 2 changes structural self weights and normalization after pair multiplicities have been added by O 1 . D and L are disabled.

Dataset	CEGCN	+O	+D	+L	+O+D+L
Actor	$71.57\pm 0.93$	$74.54\pm 1.77$	$71.58\pm 1.89$	$68.80\pm 1.46$	$75.14\pm 0.18$
Cora-CA	$60.08\pm 1.87$	$59.24\pm 2.11$	$60.21\pm 1.71$	$58.47\pm 1.58$	$60.92\pm 2.39$
DBLP-CA	$63.58\pm 2.01$	$73.87\pm 2.05$	$62.81\pm 2.60$	$63.49\pm 0.92$	$73.54\pm 0.88$
Cora	$61.82\pm 1.04$	$60.72\pm 3.32$	$61.26\pm 0.81$	$57.70\pm 2.19$	$60.71\pm 1.90$
Pubmed	$74.38\pm 0.33$	$74.57\pm 0.69$	$74.60\pm 0.29$	$73.47\pm 0.18$	$75.30\pm 1.24$
Pokec	$55.92\pm 1.60$	$54.56\pm 0.78$	$56.62\pm 0.93$	$55.93\pm 2.17$	$55.60\pm 3.13$

Appendix

Table 48: Hyperedge prediction using the identity policy for cold nodes (AUROC): original-view scores (percent), mean

\pm

sample SD over the same three seeds as the primary-metric table.

Dataset	+O	+D	+L	+O+D+L
Actor	$+2.97\pm 2.42$	$+0.01\pm 1.62$	$-2.77\pm 0.97$	$+3.57\pm 1.07$
Cora-CA	$-0.84\pm 2.97$	$+0.13\pm 0.54$	$-1.61\pm 0.63$	$+0.84\pm 3.70$
DBLP-CA	$+10.29\pm 1.03$	$-0.77\pm 0.96$	$-0.09\pm 1.89$	$+9.96\pm 1.16$
Cora	$-1.09\pm 2.68$	$-0.56\pm 1.01$	$-4.11\pm 1.43$	$-1.10\pm 1.28$
Pubmed	$+0.19\pm 0.37$	$+0.23\pm 0.17$	$-0.90\pm 0.51$	$+0.92\pm 0.96$
Pokec	$-1.36\pm 1.12$	$+0.70\pm 2.44$	$+0.01\pm 1.22$	$-0.32\pm 1.96$

Appendix

Table 49: Hyperedge prediction using the identity policy for cold nodes (AUROC): paired gains over CEGCN using the same policy for cold nodes (points), mean

\pm

sample SD over the same three seeds as the primary-metric table.

Dataset	CEGCN	+O 1	+O 1 +O 2	$\delta_{1}$	$\delta_{2}$
Actor	$71.57\pm 0.93$	$71.35\pm 1.83$	$74.54\pm 1.77$	$-0.22\pm 0.98$	$3.19\pm 2.91$
Cora-CA	$60.08\pm 1.87$	$59.64\pm 1.53$	$59.24\pm 2.11$	$-0.44\pm 0.69$	$-0.40\pm 3.11$
DBLP-CA	$63.58\pm 2.01$	$63.95\pm 2.18$	$73.87\pm 2.05$	$0.37\pm 0.40$	$9.92\pm 0.73$
Cora	$61.82\pm 1.04$	$61.08\pm 1.14$	$60.72\pm 3.32$	$-0.73\pm 1.31$	$-0.36\pm 2.40$
Pubmed	$74.38\pm 0.33$	$74.97\pm 0.17$	$74.57\pm 0.69$	$0.59\pm 0.17$	$-0.40\pm 0.52$
Pokec	$55.92\pm 1.60$	$56.53\pm 0.80$	$54.56\pm 0.78$	$0.61\pm 2.40$	$-1.97\pm 1.47$

Appendix

Table 50: Hyperedge prediction: successive changes to the propagation operator (AUROC; identity policy for cold nodes). Original-view scores (percent) and paired stepwise changes (percentage points) are mean

\pm

sample SD. O 1 adds pair multiplicities; O 2 changes structural self weights and normalization after pair multiplicities have been added by O 1 . D and L are disabled.

Dataset	CEGCN	+O	+D	+L	+O+D+L
Actor	$65.44\pm 2.43$	$65.12\pm 0.75$	$63.48\pm 0.66$	$65.09\pm 2.54$	$67.08\pm 4.28$
Cora-CA	$61.17\pm 6.87$	$53.74\pm 1.44$	$62.03\pm 6.82$	$60.59\pm 2.12$	$62.88\pm 1.11$
DBLP-CA	$65.13\pm 3.35$	$60.45\pm 1.38$	$64.86\pm 3.43$	$69.36\pm 1.03$	$69.40\pm 0.37$
Cora	$59.89\pm 0.89$	$58.58\pm 0.57$	$59.40\pm 1.22$	$58.84\pm 1.36$	$58.87\pm 1.32$
Pubmed	$72.26\pm 1.02$	$72.17\pm 0.45$	$72.47\pm 0.76$	$72.04\pm 1.05$	$73.16\pm 0.32$
Pokec	$75.22\pm 3.07$	$72.69\pm 2.70$	$73.56\pm 0.87$	$75.21\pm 1.26$	$74.13\pm 1.34$

Appendix

Table 51: Hyperedge prediction using the zero-message policy for cold nodes (AUROC): original-view scores (percent), mean

\pm

sample SD.

Dataset	+O	+D	+L	+O+D+L
Actor	$-0.32\pm 2.07$	$-1.97\pm 1.94$	$-0.35\pm 4.96$	$+1.64\pm 2.91$
Cora-CA	$-7.43\pm 6.47$	$+0.86\pm 0.28$	$-0.57\pm 6.57$	$+1.71\pm 6.66$
DBLP-CA	$-4.68\pm 3.48$	$-0.27\pm 0.13$	$+4.23\pm 2.41$	$+4.26\pm 2.98$
Cora	$-1.31\pm 1.31$	$-0.49\pm 0.80$	$-1.05\pm 1.19$	$-1.02\pm 1.98$
Pubmed	$-0.09\pm 0.58$	$+0.22\pm 0.34$	$-0.21\pm 0.41$	$+0.90\pm 0.93$
Pokec	$-2.54\pm 4.66$	$-1.66\pm 2.21$	$-0.01\pm 2.08$	$-1.09\pm 1.75$

Appendix

Table 52: Hyperedge prediction using the zero-message policy for cold nodes (AUROC): paired gains over CEGCN using the same policy for cold nodes (points), mean

\pm

sample SD.

Dataset	CEGCN	+O 1	+O 1 +O 2	$\delta_{1}$	$\delta_{2}$
Actor	$65.44\pm 2.43$	$63.07\pm 1.14$	$65.12\pm 0.75$	$-2.37\pm 1.51$	$2.05\pm 1.25$
Cora-CA	$61.17\pm 6.87$	$60.03\pm 6.36$	$53.74\pm 1.44$	$-1.13\pm 0.81$	$-6.30\pm 5.83$
DBLP-CA	$65.13\pm 3.35$	$62.03\pm 2.07$	$60.45\pm 1.38$	$-3.10\pm 1.96$	$-1.58\pm 2.96$
Cora	$59.89\pm 0.89$	$59.66\pm 1.01$	$58.58\pm 0.57$	$-0.23\pm 0.88$	$-1.08\pm 1.58$
Pubmed	$72.26\pm 1.02$	$72.86\pm 1.08$	$72.17\pm 0.45$	$0.61\pm 0.59$	$-0.70\pm 0.77$
Pokec	$75.22\pm 3.07$	$74.71\pm 1.12$	$72.69\pm 2.70$	$-0.51\pm 3.07$	$-2.02\pm 1.79$

Appendix

Table 53: Hyperedge prediction: successive changes to the propagation operator (AUROC; zero-message policy for cold nodes). Original-view scores (percent) and paired stepwise changes (percentage points) are mean

\pm

sample SD. O 1 adds pair multiplicities; O 2 changes structural self weights and normalization after pair multiplicities have been added by O 1 . D and L are disabled.

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Spectral Higher-Order Neural Networks Have Sharp Expressivity Bounds

Gianluca Peri, Diego Febbe, Duccio Fanelli

Department of Physics, University of Florence & INFN, Italy

Hypergraph Neural Networks Neural Network Expressivity

Open paper AskPDF arXiv

Neural hypergraphs are a natural generalization of neural networks, the reference models in modern machine learning. Yet, their deployment has proven demanding: the number of weighted hyperedges required leads to an intractable parameter explosion. However, a novel parametrization that leverages spectral attributes for neural hypergraphs has been recently proposed, that enables to recycle parameters via a weight sharing scheme and consequently yields a significant reduction of the associated computational cost. Preliminary tests carried out on spectral higher-order architectures pointed to meaningful improvements in both performance and interpretability. Building on these results, we advance the benchmarking efforts by evaluating the spectral higher order framework on N-bit parity tasks, a well-established testbed known to be particularly challenging. As we will convincingly argue, Spectral Higher-Order Neural Networks (SHONNs) possess a versatile and highly tunable hypothesis space.

Abstract

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