A Topological Characterization of Graph Neural Networks via Stochastic Block Model Embeddings on the n-Sphere

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 ${\text{sphere}}^{n-1}\subset {\mathbb{R}}^n$. The construction rests on three classical pillars: the \emph{compactness} of the cut-distance graphon space $(\text{Wo},\text{cutdist})$ \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 $\epsilon >0$, a trained MPNN $\mathrm{\Phi }$ acting on a sufficiently large graph factors (up to $\epsilon$) through a step-graphon-signal of bounded complexity, and we construct an explicit measure-preserving map ${\mathrm{\Psi }}_n\text{\hspace{0.17em}⁣}:[0,1]\to {\text{sphere}}^{n-1}$ 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.