EDiS: Edge Disjoint Subgraph Sparsification Framework for Graph Neural Networks

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.

Graph neural networks (GNN) based on message passing are provably no more powerful than the one-dimensional Weisfeiler--Leman colour-refinement test (1-WL): two graphs it cannot tell apart receive identical representations, however deep or wide the network. A common remedy augments node or edge features with precomputed structural descriptors, most often counts of a fixed small subgraph such as triangles or longer cycles, but such counts require committing in advance to the size of the substructure counted, a choice usually made blind to the data. We study a descriptor that avoids this choice. The edge-girth of an edge is the length of a shortest cycle through it, and its multiplicity is the number of such shortest cycles; together they form a per-edge invariant that reports cycles of arbitrary length, computable exactly by a single breadth-first search per edge. Injected into a gated message-passing architecture, EGAGNN, it reaches a test MAE a factor three below the closest gated comparator on the ZINC-12k regression benchmark at 104k parameters; against bounded cycle-counting descriptors under the same architecture, it matches only a dictionary counting cycles up to length eight, using twice as many channels, while a dictionary capped at length four performs no better than no structural information at all. On graph discrimination we prove a matching limitation: on graphs where every edge sees the same number of shortest cycles of the same length, the descriptor becomes constant and any model built on it collapses back to the 1-WL bound. This holds without exception across all 400 pairs of the BREC benchmark: not one of the 90 such pairs is distinguished.

Mini-batch training of Graph Neural Networks (GNNs) is fundamentally different from training on i.i.d. data: sampling a subgraph alters the topology and introduces boundary effects, leading prior work to develop structure-aware samplers that preserve local connectivity and reduce embedding variance. Surprisingly, we demonstrate that the simplest possible scheme, Random Node Sampling (RNS), training on the induced subgraph of uniformly sampled nodes, matches or outperforms full-graph training on 8 of 10 datasets at a fraction of the wall-clock time and memory. To explain this, we apply backward error analysis to graph mini-batch Stochastic Gradient Descent (SGD) and show that it implicitly minimizes the sampled loss plus a regularizer proportional to the mini-batch gradient variance, a quantity directly shaped by the sampler. Although RNS discards local structure, it produces mini-batches whose expected loss is closer to the full-graph loss, and whose per-batch gradients have lower variance, yielding a better implicit objective. Our analysis reframes the choice of graph sampler as a form of implicit regularization, and identifies RNS as a strong, theoretically grounded method for scalable GNN training.