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This research introduces a topological framework that maps Graph Neural Network behaviors onto an n-sphere, creating a low-dimensional fingerprint for visual inspection and cross-model search. By leveraging classical graphon theory and Lipschitz continuity, the method enables transfer-learning candidate retrieval without retraining, while explicitly addressing high-dimensional concentration of measure issues relevant to large-scale embeddings. The paper outlines future directions including hyperbolic alternatives and information-geometric reformulations to further refine these model comparison techniques.
A Topological Characterization of Graph Neural Networks via Stochastic Block Model Embeddings on the n-Sphere
arXiv:2606.07598v1 Announce Type: new Abstract: We propose a topological framework for comparing trained Graph Neural Networks (GNNs) by mapping the Stochastic Block Models (SBMs) induced on the graphon-signal space of a Message Passing Neural Network (MPNN) onto the unit $n$-sphere $\sphere^{n-1}\subset\R^n$. The construction rests on three classical pillars: the \emph{compactness} of the cut-distance graphon space $(\Wo,\cutdist)$ \citep{lovasz2006limits,lovasz2012large}, the Frieze--Kannan \emph{weak regularity lemma} together with its graphon-signal extension due to \citet{levie2023graphon}, and the Lipschitz continuity of MPNNs with respect to the cut-distance. We show that, for any prescribed tolerance $\varepsilon>0$, a trained MPNN $\Phi$ acting on a sufficiently large graph factors (up to $\varepsilon$) through a step-graphon-signal of bounded complexity, and we construct an explicit measure-preserving map $\Psi_n\colon[0,1]\to\sphere^{n-1}$ that places the SBM regions on disjoint spherical caps. This produces a problem-agnostic, low-dimensional ``fingerprint'' of a trained GNN that is amenable to visual inspection and to nearest-neighbour search across model zoos, enabling \emph{transfer-learning candidate retrieval} without retraining. We discuss the obstruction posed by concentration of measure in high dimension -- a phenomenon directly relevant to LLM-scale embeddings. We close with five concrete future research directions: hyperbolic and Grassmannian alternatives to the spherical model, Gromov--Wasserstein distances on graphon-signals as an isometry-free alternative to the $n$-sphere map, an information-geometric (Fisher) reformulation of the SBM manifold, persistent-homology fingerprints of layer-wise embedding clouds, and a spectral-distance baseline derived from the graphon eigendecomposition.
Abstract:We propose a topological framework for comparing trained Graph Neural Networks (GNNs) by mapping the Stochastic Block Models (SBMs) induced on the graphon-signal space of a Message Passing Neural Network (MPNN) onto the unit $n$-sphere $\sphere^{n-1}\subset\R^n$. The construction rests on three classical pillars: the \emph{compactness} of the cut-distance graphon space $(\Wo,\cutdist)$ \citep{lovasz2006limits,lovasz2012large}, the Frieze--Kannan \emph{weak regularity lemma} together with its graphon-signal extension due to \citet{levie2023graphon}, and the Lipschitz continuity of MPNNs with respect to the cut-distance. We show that, for any prescribed tolerance $\varepsilon>0$, a trained MPNN $\Phi$ acting on a sufficiently large graph factors (up to $\varepsilon$) through a step-graphon-signal of bounded complexity, and we construct an explicit measure-preserving map $\Psi_n\colon[0,1]\to\sphere^{n-1}$ that places the SBM regions on disjoint spherical caps. This produces a problem-agnostic, low-dimensional ``fingerprint'' of a trained GNN that is amenable to visual inspection and to nearest-neighbour search across model zoos, enabling \emph{transfer-learning candidate retrieval} without retraining. We discuss the obstruction posed by concentration of measure in high dimension -- a phenomenon directly relevant to LLM-scale embeddings. We close with five concrete future research directions: hyperbolic and Grassmannian alternatives to the spherical model, Gromov--Wasserstein distances on graphon-signals as an isometry-free alternative to the $n$-sphere map, an information-geometric (Fisher) reformulation of the SBM manifold, persistent-homology fingerprints of layer-wise embedding clouds, and a spectral-distance baseline derived from the graphon eigendecomposition.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI)
arXiv:2606.07598 [cs.LG]
(or arXiv:2606.07598v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2606.07598 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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