Organizations: University of Liverpool · Liverpool, UK · KTH Royal Institute of Technology · Digital Futures · Stockholm, Sweden
Abstract
We study unsupervised hypergraph alignment, where the goal is to infer node correspondences between two hypergraphs using only structural information, without node features, labels, seed matches, or side information. Direct higher-order formulations can represent hyperedge interactions faithfully, but they can be computationally demanding and cumbersome for non-uniform hypergraphs. Graph-reduction approaches introduce a different challenge: clique expansions keep the alignment problem on the original node set but collapse all hyperedge evidence into one pairwise graph, whereas bipartite expansions preserve incidence structure but enlarge the problem from nodes to nodes plus hyperedges. We introduce FALCON (Filtration-based hypergrAph aLignment via Cross-scale Optimal traNsport), an unsupervised optimal-transport framework for hypergraph alignment. Instead of representing each hypergraph by a single collapsed clique graph, FALCON constructs a filtration-induced sequence of clique-based co-occurrence dissimilarity matrices and jointly aligns all levels through one shared multi-scale Gromov--Wasserstein (GW) objective. The shared transport plan enforces a globally consistent node correspondence across filtration levels while avoiding the auxiliary hyperedge nodes introduced by bipartite expansion. Experiments on perturbation benchmarks derived from real-world hypergraphs show that FALCON is robust to structural noise and in almost all cases outperforms strong graph- and hypergraph-alignment baselines.
Network alignment identifies node correspondences across different networks and is a fundamental primitive in many data science applications, including social network analysis, fraud detection, and knowledge graph integration. However, state-of-the-art network alignment methods often achieve high accuracy by repeatedly constructing and updating dense matrices, sacrificing scalability in the process. To address this scalability limitation without compromising alignment accuracy, we present FastAlign, a scalable, sparsity-aware framework for optimal transport-based network alignment. Rather than introducing a new alignment model, FastAlign preserves the original OT formulation and reinterprets its computation as a set of recurring mixed sparse-dense operations. FastAlign combines sparsity-aware graph computation with domain-specific kernel fusion, including a custom SpMM kernel. Our results show that FastAlign achieves alignment quality comparable to state-of-the-art OT-based methods while substantially reducing end-to-end runtime up to 3.89x-9.45x on CPU and 2.24x-32.54x on GPU.
End-to-end Supervised Graph Prediction (SGP) requires a permutation-invariant loss to compare predicted and target graphs with arbitrary node orderings. Such losses typically involve a costly graph-matching problem. We first study three Optimal Transport relaxations of this problem and show, theoretically and empirically, that the Gromov-Wasserstein (GW) objective is the most suitable for SGP. Then, to avoid solving the resulting inner optimization for every training example, we propose to amortize the graph matching (node alignment) problem. For each training sample, the loss function leverages a transport plan provided by a parametric matcher based on the differentiable Sinkhorn algorithm applied on empirical node distributions. The graph prediction module and the matcher are jointly learned. We showcase the efficiency of this approach on toy and real world SGP problems of increasing complexity including a novel Mass-spectra to Scaffold task that we introduce.
This paper introduces Diffusion Semi-Relaxed Fused Gromov-Wasserstein (DsrFGW), a novel method for graph comparison that unifies node features and structural connectivity through optimal transport. While traditional Gromov-Wasserstein and semi-relaxed variants (srGW, srFGW) capture graph structure, they often struggle with sparse, noisy, or partially observed graphs. Inspired by Graph Diffusion Distance, which posits graphs are similar if they enable similar information transmission patterns, DsrFGW incorporates diffusion processes allowing information propagation across nodes, capturing local and global structural patterns while reducing sensitivity to noise or missing edges. An extensive evaluation on 36 synthetic pairwise graph matching tasks (easy, medium, hard) demonstrates consistent superiority over srFGW, achieving accuracy improvements of 0-20 percentage points and dramatic Adjusted Rand Index (ARI) gains: in medium-difficulty scenarios, srFGW often achieves negative ARI (worse than random) while DsrFGW offers better performance in terms of both internal and external clustering quality measures (i.e., Adjusted Rank Index and Accuracy with respect to the true underlying clusters, respectively). Even under severe noise, DsrFGW improves clustering quality in 92% of the synthetic tasks with optimal diffusion scales adapting to problem difficulty, establishing DsrFGW as a robust framework for graph comparison under structural uncertainty.