We develop a spectral theory for \emph{normalized corrected GNN propagation}. The object of study is the symmetric normalized adjacency with its degree-stationary component removed, matching the normalization used by standard GCN-style models while isolating the stationary direction most directly tied to oversmoothing. The central theoretical question is whether this corrected normalized operator preserves class-discriminative signal after many propagation layers. Our main result is a high-probability exact-recovery theorem for the binary Contextual Stochastic Block Model after k=O(logn) propagation steps in the dense polylogarithmic regime p≥ClogBn/n, for any fixed B>4, under explicit graph-signal and feature-SNR conditions. We also establish a multi-class partial recovery theorem showing contraction toward class centers for most nodes. Synthetic and real node-classification experiments are included as empirical checks of the theory's predicted dependence on depth, graph signal, and feature noise.
Graph Neural Networks (GNNs) perform computations on graphs by routing the signal between graph regions using a graph shift operator or a message passing scheme. Often, the propagation of the signal leads to a loss of information, where the signal tends to diffuse across the graph instead of being deliberately routed between regions of interest. Two notions that depict this phenomenon are oversmoothing and oversquashing. In this paper, we propose an alternative approach for modeling signal propagation, inspired by quantum mechanics, using the notion of observables. Specifically, we model the place in the graph where the signal lies, how much the signal is concentrated there, and how much of the signal is propagated towards a location of interest when applying a GNN. Using these new concepts, we prove that standard spectral GNNs have poor signal propagation capabilities. We then propose a new type of spectral GNN, termed Schrödinger GNN, which we show has a superior capacity to route the signal across the graph.
Training Graph Neural Networks on large graphs is challenged by the memory cost of storing all node representations across layers. We show that several existing scalable approaches can be written as structured modifications of the GNN propagation matrix, providing a unified perspective that exposes their respective limitations. In particular, graph coarsening replaces it by a low-rank approximation that enables spectral guarantees but assigns uniform representations to clustered nodes, while Cluster-GCN restricts the propagation matrix to intra-cluster connections that allow efficient batching but sever long-range information. These are complementary failures of the \emph{same} decomposition of the graph into groups of nodes. To obtain the best of both worlds, we propose \textbf{CoRe-GNN}, which performs both propagations in parallel at each layer: a coarsened inter-cluster term capturing long-range structure, and a local intra-cluster term preserving per-node discriminability. We prove that CoRe-GNN inherits analogous approximation guarantees to those of graph coarsening, and introduce a natural cluster-based \emph{batching scheme} that scales to graphs with millions of nodes. On node classification benchmarks spanning homophilic, heterophilic, large-scale, and long-range graphs, CoRe-GNN outperforms both graph coarsening and Cluster-GCN baselines. Notably, CoRe-GNN reaches competitive accuracy on \emph{long-range} tasks, while remaining memory-efficient through batching.
Pre-propagation graph neural networks (PPGNNs) decouple node feature propagation from transformation: graph diffusion is performed once as preprocessing, and training reduces to dense per-node transformations. This design enables mini-batch training without inter-node dependencies, avoids repeated sparse matrix--matrix multiplications, and better matches modern accelerators optimized for dense compute. However, their expressivity remains unclear, and empirical results show a gap between PPGNNs and their message-passing counterparts on commonly used graph benchmarks, especially heterophilic ones. In this paper, we propose a suite of robust graph diffusion operators for preprocessing and a few-shot hidden-state re-propagation scheme during training. Our methods improve the validation and test accuracy of PPGNNs, enabling them to match the accuracy of message-passing GNNs while maintaining training efficiency.