Concorde: Geometry-Aware Link Prediction via Decoupled Energy Minimization

Link prediction remains difficult in sparse graphs, where diffusion heuristics lack connectivity and geometric methods suffer spectral instability when coupling learned representations with diffusion operators. We introduce **Concorde**, an energy-based framework that resolves this by decomposing link likelihood into three structurally decoupled branches: **(i)** a parameter-free spectral anchor that establishes permutation-invariant topological stability; **(ii)** a mixed-curvature residual on a learnable $\mathbb{H}\_{c\_h}^{d\_h} \times \mathbb{S}\_{c\_s}^{d\_s}$ product manifold, regularized by a feature-aware Ollivier--Ricci curvature proxy; and **(iii)** a continuous vector field on $\mathcal{T}\mathbb{H}\_c^d$ for non-local structural alignment across topological gaps. The branches are fused via learned gating and trained with an $N$-pair ranking loss aligned with retrieval-style evaluation. Across five benchmarks spanning sparse, dense, featureless, and attributed regimes, **Concorde** establishes state-of-the-art early retrieval precision, nearly doubling $P@10\\%$ over leading baselines in sparse settings.