This work develops this perspective theoretically, showing that quotient couplings can be lifted to aligned representatives without additional cost and that symmetrization yields equivariant flow-matching minimizers, including for categorical endpoint prediction.
Abstract
Graphs are invariant under node permutations, motivating the use of permutation-equivariant architectures in generative models. In flow matching, however, symmetry may also enter the source--target coupling: once graph pairs are compared up to node relabeling, the natural Wasserstein geometry is that of the graph quotient space. The Euclidean quotient metric of this space coincides with the Gromov--Monge distance, obtained by optimally relabeling the nodes. We develop this perspective theoretically, showing that quotient couplings can be lifted to aligned representatives without additional cost and that symmetrization yields equivariant flow-matching minimizers, including for categorical endpoint prediction. In practice, exact Gromov--Monge alignment is intractable, so we construct minibatch couplings using efficient Gromov--Wasserstein-type relaxations and lower bounds for the inner node alignment, optionally combined with an outer assignment between graphs. The resulting procedure changes only the training coupling and is compatible with standard permutation-equivariant architectures. Across continuous graph and categorical molecular generation, these structure-aware couplings substantially improve sample quality at small integration budgets, while our scaled-up molecular models remain competitive under conventional many-step sampling.
This approach demonstrates that models trained on small-scale random graphs learn to extract universal distance-preserving features, achieving robust generalization to large-scale, real-world networks that match or exceed the fidelity of classical, exact landmark-based embeddings.
Equivariant graph neural networks have proven effective tools for inference of material's properties directly from their structure. Traditionally, these have been applied such that they respect full $O(3)$ equivariance, so that any rotation or reflection of the input structure is respected in the model's output. While this works for general arrangements of atoms, additional symmetries of atomistic systems are left unleveraged. Furthermore, any symmetries of the filter functions are implicitly learned from the full dataset and not strictly enforced. In this work, we introduce point-group symmetry aware equivariant graph neural networks (PGEqNN) for materials science, with filter functions aligned with symmetry-aware indices for greater granularity in predictive tasks. With this architecture, we show that most of the predictive power of equivariant networks for tensorial elastic and dielectric datasets lies in the trivial subspaces of the point-group adapted bases. Exploiting this, an $A_1$-restricted variant matches or improves on its full point-group and $SO(3)$-partitioned counterparts while training fewer active parameters, yielding leaner models of equal accuracy.
This article proposes novel graph kernels based on quantum Rényi $\alpha $ -entropies of different orders, computed from both the unnormalized and normalized Laplacian matrices, and demonstrates that these methods achieve competitive or superior performance compared with state-of-the-art techniques, including deep learning approaches, while remaining computationally efficient.
Furqan Aziz· IEEE Transactions on Neural...· 0 citations
Efficiently and robustly analyzing shape data is critical across many scientific disciplines. While chirality is a fundamental property in numerous applications - most notably in molecular science - existing shape analysis metrics fail to distinguish between a shape and its mirror image. To address this gap, we introduce a multilinear generalization of the Gromov-Wasserstein objective. Under mild assumptions, this objective yields a distance between shapes, represented as probability distributions quotiented by a symmetry group $G$. In particular, for $G = SO(d)$, we introduce the Chiral Gromov-Wasserstein ($\mathrm{CGW}$) distance, sensitive to chirality. We establish robustness properties for the multilinear Gromov-Wasserstein distances and develop efficient algorithms to compute them, reformulating the underlying optimization problem by projecting couplings onto a low-dimensional space. We derive algorithms for both local and approximate global solutions, yielding a fully polynomial-time approximation scheme for these problems. We validate the framework through numerical experiments that demonstrate the effectiveness of $\mathrm{CGW}$ as a shape metric for chiral objects.
Clément Soubrier, Geoffrey Woollard, Andrew Warren et al.· 0 citations
We study the Equivalent Local Sequence Problem (ELSP) for simple undirected graphs using Bouchet’s isotropic-system formalism and normal matrices over \(\mathbb F_2\). Although Bouchet’s theory characterizes graph local equivalence in polynomial time, converting normal-matrix certificates into explicit graph transformations remains challenging. We introduce the Normal-Matrix Factorization Problem (NMFP), which asks whether a normal-matrix witness of local equivalence has a graph-compatible factorization into elementary transformations. Whenever such a factorization exists, an explicit local-complementation sequence can be recovered in polynomial time. Thus, the constructive part of ELSP reduces to NMFP, identifying normal-matrix factorization as its main unresolved algebraic difficulty. We apply this framework to undirected Paley graphs. In contrast to the directed case, whose local-complementation dynamics are abelian and admit linear inversion, the undirected case is noncommutative and has a more intricate stabilizer structure. Using normal matrices, we analyze Paley-graph orbits and stabilizers, derive algebraic constraints on stabilizing transformations, and completely verify the first undirected Paley graph \(P_5\). These results establish NMFP as central to constructive local equivalence and reveal connections among isotropic systems, graph transformations, and algebraic stabilizers.
Zhour Oumazouz· Journal of Combinatorial Mat...· 0 citations
Finite groups are rigid algebraic objects, whose Cayley graphs expose a rich network geometry through which group-theoretic structure can be measured, compared, and learned. In this paper, a dataset of $131{,}406$ Cayley graphs is constructed, covering all groups of order at most $767$ except order $512$, recording exact algebraic labels for group properties together with a broad collection of graph, cycle, distance, and spectral statistics. This census aims to provide novel benchmarks for studying how finite-group properties are reflected in Cayley graph observables. It also yields new enumerative contributions: alongside recovering known OEIS sequences for standard group classes, new sequences for monolithic groups and for groups generated by at most three, four, and five elements are contributed to the OEIS. The accompanying network analysis identifies several empirical regularities and formulates testable conjectures, including relationships involving square clustering, Cayley graph diameter, average graph disorder, and spectral eigengaps of nilpotent groups. Finally, a comparison between classical models, an MLP, and graph neural network architectures is performed for predicting algebraic group properties directly from Cayley graph data. The results show that engineered graph statistics are highly informative, while GNNs, especially GIN and in some fixed-order settings GCN, can recover substantial structural signal directly from the graph. Such that graph-aware architectures show phases of optimality on these group-theoretic graph representations.
Rashid Barket, Enrico Grimaldi, Yacoub Hendi et al.· 0 citations