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 spheren-1subsetRn. The construction rests on three classical pillars: the compactness of the cut-distance graphon space (Wo,cutdist) citeplovasz2006limits,lovasz2012large, the Frieze--Kannan weak regularity lemma together with its graphon-signal extension due to citetlevie2023graphon, 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 Psincolon[0,1]→spheren-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 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.