Unsupervised Network Embedding Beyond Homophily

Zhiqiang Zhong, Guadalupe Gonzalez, Daniele Grattarola, Jun Pang

Abstract

Network embedding (NE) approaches have emerged as a predominant technique to represent complex networks and have benefited numerous tasks. However, most NE approaches rely on a homophily assumption to learn embeddings with the guidance of supervisory signals, leaving the unsupervised heterophilous scenario relatively unexplored. This problem becomes especially relevant in fields where a scarcity of labels exists. Here, we formulate the unsupervised NE task as an $r$-ego network discrimination problem and develop the SELENE framework for learning on networks with homophily and heterophily. Specifically, we design a dual-channel feature embedding pipeline to discriminate $r$-ego networks using node attributes and structural information separately. We employ heterophily adapted self-supervised learning objective functions to optimise the framework to learn intrinsic node embeddings. We show that SELENE's components improve the quality of node embeddings, facilitating the discrimination of connected heterophilous nodes. Comprehensive empirical evaluations on both synthetic and real-world datasets with varying homophily ratios validate the effectiveness of SELENE in homophilous and heterophilous settings showing an up to $12.52\

Benchmark Results

Tested on 3 datasets 4 baseline values reported too low Non-standard metric on 2 datasets

Tables show presented results in global context (best 3 papers + GCN + MLP).

Heterophilic Graphs

Texas

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Default metric is Accuracy. The paper reports 0 values on Accuracy , 13 values on ACC , 13 values on F1 and 13 values on ROC-AUC .

Sorted by ACC. Highlighted rows are from other papers.

#	Model	Accuracy ↑ default	ACC ↑	F1 ↑	ROC-AUC ↑	Paper
1	SELENE Hyb ★ proposed	—	0.6523	—	—	Accuracy — F1 — ROC-AUC — This paper
2	NE Hyb ★ proposed	—	—	0.6430	0.7840	Accuracy — F1 0.6430 ROC-AUC 0.7840 This paper
3	FAGCN GNN baseline	—	0.5792	0.6070	0.6730	Accuracy — F1 0.6070 ROC-AUC 0.6730 This paper
5	SAGE GNN baseline	—	0.5683	0.5630	0.5590	Accuracy — F1 0.5630 ROC-AUC 0.5590 This paper
4	GPRGNN GNN baseline	—	0.5750	0.5610	0.7820	Accuracy — F1 0.5610 ROC-AUC 0.7820 This paper
6	DGI GNN baseline	—	0.5574	0.5410	0.7870	Accuracy — F1 0.5410 ROC-AUC 0.7870 This paper
7	GBT GNN baseline	—	0.5546	0.4610	0.7830	Accuracy — F1 0.4610 ROC-AUC 0.7830 This paper
2	H2GCN GNN baseline	—	0.5880	0.5230	0.4510	Accuracy — F1 0.5230 ROC-AUC 0.4510 This paper
8	AE Walk baseline	—	0.5049	0.5980	0.5290	Accuracy — F1 0.5980 ROC-AUC 0.5290 This paper
9	S2V GNN baseline	—	0.4973	0.5920	0.5710	Accuracy — F1 0.5920 ROC-AUC 0.5710 This paper
10	N2V Walk baseline	—	0.4880	0.5090	0.5490	Accuracy — F1 0.5090 ROC-AUC 0.5490 This paper
11	SDCN GNN baseline	—	0.4404	0.5520	0.6400	Accuracy — F1 0.5520 ROC-AUC 0.6400 This paper
12	GAE GNN baseline	—	0.4202	0.5000	0.6340	Accuracy — F1 0.5000 ROC-AUC 0.6340 This paper
13	GMI GNN baseline	—	0.3519	0.5050	0.7770	Accuracy — F1 0.5050 ROC-AUC 0.7770 This paper

Actor

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Default metric is Accuracy. The paper reports 6 values on Accuracy , 1 value on ACC (node clustering accuracy) , 13 values on F1 , 13 values on ROC-AUC and 6 values on ACC (clustering accuracy) .

#	Model	Accuracy ↑ default	ACC (node clustering accuracy) ↑	F1 ↑	ROC-AUC ↑	ACC (clustering accuracy) ↑	Paper
1	DiGGR Hyb global	0.4535 ± 0.0353	—	—	—	—	ACC (node clustering accuracy) — F1 — ROC-AUC — ACC (clustering accuracy) — Disentangled Generative Graph Representation Learning
2	TRIGON code ↗ Hyb global	0.4402 ± 0.0028	—	—	—	—	ACC (node clustering accuracy) — F1 — ROC-AUC — ACC (clustering accuracy) — Dynamic Triangulation-Based Graph Rewiring for Graph Neural Networks code ↗
3	DoG code ↗ ext data GNN global	0.4320 ± 0.0120	—	—	—	—	ACC (node clustering accuracy) — F1 — ROC-AUC — ACC (clustering accuracy) — Diffusion on Graph: Augmentation of Graph Structure for Node Classification code ↗
42	MLP Trad global	0.4026 ± 0.0108	—	—	—	—	ACC (node clustering accuracy) — F1 — ROC-AUC — ACC (clustering accuracy) — Heterophily-Aware Graph Attention Network
52	GPRGNN GNN baseline	0.3992 ±0.0067 best reported	—	0.2840	0.7960	—	ACC (node clustering accuracy) — F1 0.2840 ROC-AUC 0.7960 ACC (clustering accuracy) — paper 0.2580 best at pub: Simplifying approach to Node Classification in Graph Neural Networks (2021) best today: GrokFormer: Graph Fourier Kolmogorov-Arnold Transformers (ICML 2024) paper 0.2580 Δ at pub 0.1024 Δ today 0.1412
56	FAGCN GNN baseline	0.3970 best reported	—	0.2810	0.6130	—	ACC (node clustering accuracy) — F1 0.2810 ROC-AUC 0.6130 ACC (clustering accuracy) — paper 0.2561 best at pub: Beyond Low-frequency Information in Graph Convolutional Networks (AAAI 2021) paper 0.2561 Δ at pub 0.1409
68	GraphSAGE code ↗ GNN global	0.3899 ± 0.0085	—	—	—	—	ACC (node clustering accuracy) — F1 — ROC-AUC — ACC (clustering accuracy) — A Quasi-Wasserstein Loss for Learning Graph Neural Networks code ↗
71	H2GCN GNN baseline	0.3885 ±0.0117 best reported	—	0.2440	0.6790	—	ACC (node clustering accuracy) — F1 0.2440 ROC-AUC 0.6790 ACC (clustering accuracy) — paper 0.2555 best at pub: Graph Neural Networks Inspired by Classical Iterative Algorithms (ICML 2021) best today: $P^2$GNN: Two Prototype Sets to boost GNN Performance (2026) paper 0.2555 Δ at pub 0.1031 Δ today 0.1330
268	NE Hyb ★ proposed	—	—	0.3410	0.7250	—	ACC (node clustering accuracy) — F1 0.3410 ROC-AUC 0.7250 ACC (clustering accuracy) — This paper
335	DGI GNN baseline	0.3067 ±0.0063 best reported	—	0.2720	0.6260	—	ACC (node clustering accuracy) — F1 0.2720 ROC-AUC 0.6260 ACC (clustering accuracy) — paper 0.2426 best at pub: Graph Representation Learning via Aggregation Enhancement (2022) best today: Toward General Digraph Contrastive Learning: A Dual Spatial Perspective (2025) paper 0.2426 Δ at pub 0.0294 Δ today 0.0641
267	SELENE Hyb ★ proposed	—	0.2903	—	—	—	ACC (node clustering accuracy) 0.2903 F1 — ROC-AUC — ACC (clustering accuracy) — This paper
399	GMI GNN baseline	0.2782 best reported	—	0.2720	0.6090	—	ACC (node clustering accuracy) — F1 0.2720 ROC-AUC 0.6090 ACC (clustering accuracy) — paper 0.2618 best today: Beyond Smoothing: Unsupervised Graph Representation Learning with Edge Heterophily Discriminating (AAAI 2022) paper 0.2618 Δ today 0.0164
436	N2V Walk baseline	—	—	0.2400	0.6730	0.2502	ACC (node clustering accuracy) — F1 0.2400 ROC-AUC 0.6730 ACC (clustering accuracy) 0.2502 This paper
438	GBT GNN baseline	0.2468	—	0.2450	0.5980	—	ACC (node clustering accuracy) — F1 0.2450 ROC-AUC 0.5980 ACC (clustering accuracy) — This paper
441	AE Walk baseline	—	—	0.2840	0.5010	0.2419	ACC (node clustering accuracy) — F1 0.2840 ROC-AUC 0.5010 ACC (clustering accuracy) 0.2419 This paper
450	SDCN GNN baseline	—	—	0.2520	0.5520	0.2367	ACC (node clustering accuracy) — F1 0.2520 ROC-AUC 0.5520 ACC (clustering accuracy) 0.2367 This paper
383	GAE GNN baseline	—	—	0.2540	0.6680	0.2345	ACC (node clustering accuracy) — F1 0.2540 ROC-AUC 0.6680 ACC (clustering accuracy) 0.2345 This paper
454	SAGE GNN baseline	—	—	0.2770	0.6340	0.2308	ACC (node clustering accuracy) — F1 0.2770 ROC-AUC 0.6340 ACC (clustering accuracy) 0.2308 This paper
463	S2V GNN baseline	—	—	0.2270	0.6520	0.2249	ACC (node clustering accuracy) — F1 0.2270 ROC-AUC 0.6520 ACC (clustering accuracy) 0.2249 This paper

Heterogeneous Graph Benchmarks

DBLP

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Default metric is Micro-F1. The paper reports 0 values on Micro-F1 , 13 values on ACC , 13 values on NMI , 13 values on ARI , 13 values on F1 and 7 values on ROC-AUC .

Sorted by ACC. Highlighted rows are from other papers.

#	Model	Micro-F1 ↑ default	ACC ↑	NMI ↑	ARI ↑	F1 ↑	ROC-AUC ↑	Paper
1	DANE-O Trad global	0.7721	—	—	—	—	—	Micro-F1 0.7721 NMI — ARI — F1 — ROC-AUC — Attributed Network Embedding for Learning in a Dynamic Environment
2	DANE Trad global	0.7664	—	—	—	—	—	Micro-F1 0.7664 NMI — ARI — F1 — ROC-AUC — Attributed Network Embedding for Learning in a Dynamic Environment
3	SELENE Hyb ★ proposed	—	0.7574	0.4417	—	—	—	Micro-F1 — NMI 0.4417 ARI — F1 — ROC-AUC — This paper
6	GBT GNN baseline	—	0.7310	0.4221	0.4257	0.7440	—	Micro-F1 — NMI 0.4221 ARI 0.4257 F1 0.7440 ROC-AUC — This paper
15	GMI GNN baseline	—	0.6328	0.3391	0.2877	0.8110	—	Micro-F1 — NMI 0.3391 ARI 0.2877 F1 0.8110 ROC-AUC — This paper
16	SDCN GNN baseline	—	0.6194	0.2713	0.2777	0.6380	0.6280	Micro-F1 — NMI 0.2713 ARI 0.2777 F1 0.6380 ROC-AUC 0.6280 This paper
17	DGI GNN baseline	—	0.5822	0.2998	0.2681	0.7920	0.9390	Micro-F1 — NMI 0.2998 ARI 0.2681 F1 0.7920 ROC-AUC 0.9390 This paper
18	GAE GNN baseline	—	0.5781	0.2894	0.1878	0.7680	0.8700	Micro-F1 — NMI 0.2894 ARI 0.1878 F1 0.7680 ROC-AUC 0.8700 This paper
20	SAGE GNN baseline	—	0.4868	0.1646	0.1338	0.7320	0.8410	Micro-F1 — NMI 0.1646 ARI 0.1338 F1 0.7320 ROC-AUC 0.8410 This paper
21	AE Walk baseline	—	0.4850	0.1898	0.1515	0.7460	0.8720	Micro-F1 — NMI 0.1898 ARI 0.1515 F1 0.7460 ROC-AUC 0.8720 This paper
22	NE Hyb ★ proposed	—	—	—	0.4635	0.8130	—	Micro-F1 — NMI — ARI 0.4635 F1 0.8130 ROC-AUC — This paper
23	GPRGNN* GNN baseline	—	0.4396	0.1127	0.0492	0.6430	—	Micro-F1 — NMI 0.1127 ARI 0.0492 F1 0.6430 ROC-AUC — This paper
24	H2GCN* GNN baseline	—	0.4187	0.1104	0.0467	0.4130	—	Micro-F1 — NMI 0.1104 ARI 0.0467 F1 0.4130 ROC-AUC — This paper
25	FAGCN* GNN baseline	—	0.4125	0.1060	0.0432	0.5600	—	Micro-F1 — NMI 0.1060 ARI 0.0432 F1 0.5600 ROC-AUC — This paper
26	S2V GNN baseline	—	0.3165	0.0133	0.0139	0.3340	0.7170	Micro-F1 — NMI 0.0133 ARI 0.0139 F1 0.3340 ROC-AUC 0.7170 This paper
27	N2V Walk baseline	—	0.2919	0.0014	-0.0400	0.2690	0.7580	Micro-F1 — NMI 0.0014 ARI -0.0400 F1 0.2690 ROC-AUC 0.7580 This paper

SOTA performance (not significantly different from the best) Proposed method (this paper) Baseline / comparison (this paper) Global context (other papers) Best at pub — reported result worse than the best known when published (best reported in green) Best today — OK at publication; today's leaderboard best is elsewhere