Learning on large graphs presents significant challenges, with traditional Message Passing Neural Networks suffering from computational and memory costs scaling linearly with the number of edges. We introduce the Intersecting Block Graph (IBG), a low-rank factorization of large directed graphs based on combinations of intersecting bipartite components, each consisting of a pair of communities, for source and target nodes. By giving less weight to non-edges, we show how to efficiently approximate any graph, sparse or dense, by a dense IBG. Specifically, we prove a constructive version of the weak regularity lemma, showing that for any chosen accuracy, every graph, regardless of its size or sparsity, can be approximated by a dense IBG whose rank depends only on the accuracy. This dependence of the rank solely on the accuracy, and not on the sparsity level, is in contrast to previous forms of the weak regularity lemma. We present a graph neural network architecture operating on the IBG representation of the graph and demonstrating competitive performance on node classification, spatio-temporal graph analysis, and knowledge graph completion, while having memory and computational complexity linear in the number of nodes rather than edges.

翻译：大规模图学习面临显著挑战，传统消息传递神经网络的计算与内存成本随边数线性增长。本文提出相交分块图，这是一种基于相交二分分量组合的大规模有向图低秩分解方法，每个分量由源节点与目标节点构成的一对社区组成。通过降低非边权重，我们证明了如何通过稠密IBG高效逼近任意稀疏或稠密图。具体而言，我们给出了弱正则引理的可构造版本，证明对于任意选定精度，无论图的规模或稀疏度如何，均可由秩仅取决于精度的稠密IBG逼近。这种秩仅依赖于精度而与稀疏度无关的特性，与以往弱正则引理形式形成鲜明对比。我们提出了一种在图的IBG表示上运行的图神经网络架构，在节点分类、时空图分析和知识图谱补全任务中展现出具有竞争力的性能，同时保持内存和计算复杂度与节点数而非边数呈线性关系。