Moment-guided edge sampling

Edge sampling makes local decisions to achieve graph-level objectives, such as preserving structural properties. This creates a fundamental challenge: textithow can the effect of a local edge edit (i.e., edge addition or removal) on global graph structure be quantified and controlled? We address this challenge with a textitmoment-guided edge sampling framework based on spectral moments of the random-walk transition matrix. We compute exact moment changes through two complementary methods: a combinatorial method with closed-form updates for low-order moments, and a low-rank method that exploits textitlocality and textitcyclic trace invariance to compress computations to edited endpoints, supporting arbitrary moment orders and batched edits. For single-edge edits at fixed moment orders, the low-rank method reduces the cost from O(mn) to O(m), while the combinatorial method evaluates low-order changes in constant time given maintained local statistics. These moment changes provide interpretable structural signatures of local edge motifs that aggregate into graph-level fingerprints. This structural meaning motivates us to ask whether preserving moments also preserves the graph properties. We further derive and validate that moment-preserving sampling can retain related structural properties, including triangle-weighted clustering coefficient. These structural insights enable analysis and improvement of graph learning: different edge structures have distinct effects on supervised node classification, while moment-guided augmentation is competitive for graph contrastive learning. Together, these findings establish moments as an interpretable and controllable bridge from local edge edits to global graph structure and learning.