This work presents inGRASS, a novel algorithm designed for incremental spectral sparsification of large undirected graphs. The proposed inGRASS algorithm is highly scalable and parallel-friendly, having a nearly-linear time complexity for the setup phase and the ability to update the spectral sparsifier in $O(\log N)$ time for each incremental change made to the original graph with $N$ nodes. A key component in the setup phase of inGRASS is a multilevel resistance embedding framework introduced for efficiently identifying spectrally-critical edges and effectively detecting redundant ones, which is achieved by decomposing the initial sparsifier into many node clusters with bounded effective-resistance diameters leveraging a low-resistance-diameter decomposition (LRD) scheme. The update phase of inGRASS exploits low-dimensional node embedding vectors for efficiently estimating the importance and uniqueness of each newly added edge. As demonstrated through extensive experiments, inGRASS achieves up to over $200 \times$ speedups while retaining comparable solution quality in incremental spectral sparsification of graphs obtained from various datasets, such as circuit simulations, finite element analysis, and social networks.

翻译：本文提出inGRASS算法，一种专为大规模无向图增量式谱稀疏化设计的新型算法。该算法具有高度可扩展性和并行友好性，其初始化阶段的时间复杂度接近线性，且对于包含N个节点的原始图的每次增量变化，能够以O(log N)时间更新谱稀疏器。inGRASS初始化阶段的核心是一个多级电阻嵌入框架，通过低电阻直径分解（LRD）方案将初始稀疏器分解为多个具有有界有效电阻直径的节点簇，从而高效识别谱关键边并有效检测冗余边。其更新阶段利用低维节点嵌入向量，高效估计每条新增边的重要性和独特性。通过大量实验证明，inGRASS在保持相当求解精度的前提下，对来自电路仿真、有限元分析及社交网络等数据集的图进行增量谱稀疏化时，可实现高达200倍以上的加速比。