We propose refined GRFs (GRFs++), a new class of Graph Random Features (GRFs) for efficient and accurate computations involving kernels defined on the nodes of a graph. GRFs++ resolve some of the long-standing limitations of regular GRFs, including difficulty modeling relationships between more distant nodes. They reduce dependence on sampling long graph random walks via a novel walk-stitching technique, concatenating several shorter walks without breaking unbiasedness. By applying these techniques, GRFs++ inherit the approximation quality provided by longer walks but with greater efficiency, trading sequential, inefficient sampling of a long walk for parallel computation of short walks and matrix-matrix multiplication. Furthermore, GRFs++ extend the simplistic GRFs walk termination mechanism (Bernoulli schemes with fixed halting probabilities) to a broader class of strategies, applying general distributions on the walks' lengths. This improves the approximation accuracy of graph kernels, without incurring extra computational cost. We provide empirical evaluations to showcase all our claims and complement our results with theoretical analysis.
We investigate message-passing graph neural networks with random node features. Random node features are known to enhance the expressiveness of graph neural networks (GNNs) both theoretically and empirically. Here, we establish a novel universality result focusing on permutation-equivariant neural networks (PENNs), a class of GNNs built from feedforward neural network components that subsumes many prominent GNN architectures. We show that PENNs, combined with partially random node features, can approximate arbitrarily well in probability any measurable permutation-invariant or permutation-equivariant function on directed graphs of fixed size with multidimensional node and edge features. For k-times continuously differentiable functions, k≥2, we also derive upper bounds on the approximation rates, relating the complexity of the feedforward components of a PENN in terms of layer depth and number of nonzero weights to the desired approximation accuracy.
Graph foundation models (GFMs), pretrained on massive graph data, have transformed graph machine learning by supporting general-purpose reasoning across diverse graph tasks and domains. Existing GFMs pretrained with fixed-hop subgraph sampling impose a fixed receptive field, causing scale mismatch on diverse tasks, which often require heterogeneous and unknown structural contexts beyond a fixed sampling scale. We propose R-GFM, a Riemannian Graph-of-Graphs (GoG) based foundation model, that treats structural scale as a first-class citizen in modeling. R-GFM constructs a multi-scale GoG over-sampled subgraphs at different hop distances and learns geometry-adaptive representations from Riemannian manifolds. Theoretical analysis shows that R-GFM reduces structural domain generalization error compared to fixed-scale GFMs. Experiments on various datasets demonstrate that R-GFM achieves state-of-the-art performance, with up to a 49% relative improvement on downstream tasks. Our code is available at https://github.com/USTC-DataDarknessLab/R-GFM.
Subgraph sampling reduces the training cost of large-scale graph neural networks, but sampling criteria may overlook the geometric roles of edges. We propose a resistance-curvature-guided sampling framework built on ERC-LG, a curvature approximation method for large-scale graphs. ERC-LG combines Johnson-Lindenstrauss projections with regularized multi-GPU batched conjugate gradient solvers, avoiding explicit Laplacian pseudoinverse computation and full embedding storage. The resulting curvature informs node- and edge-sampling probabilities for constructing GNN training subgraphs. Experiments show numerical agreement with pseudoinverse-based curvature and reduced runtime compared with CG-only computation. ERC-LG-based sampling variants achieve the highest mean accuracy on six of seven real-world datasets in downstream node classification.