  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.
