Knowledge graphs have emerged as fundamental structures for representing complex relational data across scientific and enterprise domains. However, existing embedding methods face critical limitations when modeling diverse relationship types at scale: Euclidean models struggle with hierarchies, vector space models cannot capture asymmetry, and hyperbolic models fail on symmetric relations. We propose HyperComplEx, a hybrid embedding framework that adaptively combines hyperbolic, complex, and Euclidean spaces via learned attention mechanisms. A relation-specific space weighting strategy dynamically selects optimal geometries for each relation type, while a multi-space consistency loss ensures coherent predictions across spaces. We evaluate HyperComplEx on computer science research knowledge graphs ranging from 1K papers (~25K triples) to 10M papers (~45M triples), demonstrating consistent improvements over state-of-the-art baselines including TransE, RotatE, DistMult, ComplEx, SEPA, and UltraE. Additional tests on standard benchmarks confirm significantly higher results than all baselines. On the 10M-paper dataset, HyperComplEx achieves 0.612 MRR, a 4.8% relative gain over the best baseline, while maintaining efficient training, achieving 85 ms inference per triple. The model scales near-linearly with graph size through adaptive dimension allocation. We release our implementation and dataset family to facilitate reproducible research in scalable knowledge graph embeddings.
Traditional knowledge graph embedding approaches assume fixed geometries—Euclidean for translation, complex for asymmetry, or hyperbolic for hierarchies—limiting their effectiveness on heterogeneous graphs that combine hierarchical, symmetric, and asymmetric relations. Scientific knowledge graphs, in particular, contain diverse relation types such as citations, authorships, collaborations, and hierarchical classifications. These constraints motivate the need for a framework that can adaptively select geometric spaces for each relation type while remaining scalable to millions of entities.
HyperComplEx represents entities and relations in a unified multi-space embedding combining hyperbolic, complex, and Euclidean geometries. Each entity e and relation r is encoded as a triplet of embeddings, capturing hierarchical, asymmetric, and translational patterns simultaneously. Subspace-specific scoring functions ϕH, ϕC, and ϕE ensure each geometry contributes appropriately to link prediction.
A relation-specific adaptive attention vector αr dynamically weights each subspace, enabling automatic geometry selection per relation. Training optimizes a combined loss L = Lrank + λ1Lconsistency + λ2Lreg, which includes a self-adversarial ranking term, multi-space consistency regularization to align embeddings across geometries, and standard ℓ2 regularization to reduce overfitting. This ensures both discriminative and geometrically coherent representations.
Scalability is achieved through adaptive dimension allocation, sharded and cached embeddings for large graphs, and mixed-precision optimization. The inference complexity scales linearly with effective embedding size, achieving sub-100 ms latency per triple even on 10M-entity graphs.
Across eight datasets, HyperComplEx consistently outperforms six strong baselines, achieving highest MRR scores (0.612 on CS-10M, 0.411 on DBP15K). Its adaptive multi-space embeddings capture diverse relational structures: hyperbolic for hierarchical relations, complex for asymmetric dependencies, and Euclidean for symmetric collaborations.
Scalability experiments show near-linear training (β1 = 1.06) and sublinear inference scaling (λ ≈ 0.42). Ablation studies confirm that both adaptive attention and multi-space consistency are critical, with largest performance degradation occurring when removing the complex subspace. These findings demonstrate that HyperComplEx provides state-of-the-art link prediction, interpretable geometric reasoning, and efficient large-scale deployment.