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Non-Euclidean Mixture Model for Social Network Embedding
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Proposes NMM, a graph formation model mixing homophily (spherical space) and social influence (hyperbolic space) link probabilities, and NMM-GNN, a non-Euclidean GraphVAE that unifies the two spaces with a space unification loss.
Problem
Social network embedding methods model link formation with a single factor in Euclidean space, missing the joint effects of homophily and social influence and the cyclic and hierarchical topologies they produce.
Approach
Models link probability as a learned mixture of a homophily term in spherical space and a social-influence term in hyperbolic space, with a projection-based space unification loss aligning the two embeddings of each node, and a non-Euclidean GNN-based VAE for encoding.
Key results
- On BlogCatalog, NMM-GNN reaches 78.8 F1 and 96.9 AUC versus 65.6 F1 and 95.1 AUC for the best prior mixture baseline RaRE.
- On LiveJournal, NMM-GNN achieves 86.8 AUC versus 79.9 for RaRE.
- On Friendster, NMM-GNN achieves 94.9 AUC and 57.7 F1, the best among reported methods.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add a non-Euclidean GNN encoder and GraphVAE framework to NMM | NMM · BlogCatalog classification | F1: 70.9 -> 78.8 | Helped |
| add a non-Euclidean GNN encoder and GraphVAE framework to NMM | NMM · LiveJournal classification | AUC: 84.2 -> 86.8 | Helped |
| add a non-Euclidean GNN encoder and GraphVAE framework to NMM | NMM · Friendster classification | JI: 79.8 -> 83.3 | Helped |
| use a mixture of homophily and social influence rather than either component alone (ablation) | NMM · all datasets (link prediction) | AUC | Helped |
| use spherical space for homophily and hyperbolic space for social influence (ablation) | NMM · LiveJournal classification | JI: 76.5 vs. 76.2 | Helped |
| add space unification loss (ablation) | NMM-GNN · Wikipedia Clickstream, BlogCatalog, LiveJournal, Wikipedia Hyperlink, Friendster | AUC | Helped |
| use non-Euclidean GNN encoders and GraphVAE training for inductive link prediction (ablation) | NMM-GNN vs RARE · LiveJournal unseen-node link prediction | AUC | Helped |
Surprising findings
- Space alignment is described as necessary because without it the hyperbolic embedding has too many degrees of freedom; ablations show the space unification loss improves AUC across datasets.
- On Wikipedia Hyperlink, NMM-GNN's AUC (97.3) is below NMM (98.0) and RaRE (97.5) is competitive, suggesting gains are not uniform across datasets.
Limitations
- GraphVAE training is less efficient than plain GNN training.
- Curvatures are fixed (K_S=1, K_H=-1) rather than learned.
- Evaluated only on featureless graphs in the main text; attributed graph experiments are limited to two datasets.
Experimental setup
Datasets: BlogCatalog LiveJournal Friendster Wikipedia Clickstream Wikipedia Hyperlink Facebook Google+
Benchmarks: BlogCatalog LiveJournal Friendster Wikipedia Clickstream Wikipedia Hyperlink Facebook Google+
Metrics: Jaccard Index Hamming Loss F1 score AUC
Baselines: GraRep RolX GraphWave GraphSAGE GCN GAT GIN GraphCL GELTOR NRP HGCN κ-GCN
Builds on: RaRE HGCN GraphVAE NRP
Closest papers by meaning
- Embedding Heterogeneous Networks into Hyperbolic Space Without Meta-path (AAAI 2021 · similarity 0.87)
- Link Prediction in Multilayer Networks via Cross-Network Embedding (AAAI 2024 · similarity 0.86)
- Bridging Theory and Practice in Link Representation with Graph Neural Networks (NeurIPS 2025 · similarity 0.86)
- On the Impact of Feature Heterophily on Link Prediction with Graph Neural Networks (NeurIPS 2024 · similarity 0.86)
- DAG-Aware Variational Autoencoder for Social Propagation Graph Generation (AAAI 2024 · similarity 0.86)
- Information Loss and Disparate Effects in Network Embeddings (NeurIPS 2026 · similarity 0.86)
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