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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(logN) 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× 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.
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Towards Scalable Spectral Embedding and Data Visualization via Spectral Coarsening
This paper proposes a scalable multilevel framework for the spectral embedding of large undirected graphs. The proposed method first computes much smaller yet sparse graphs while preserving the key spectral (structural) properties of the original graph, by exploiting a nearly-linear time spectral graph coarsening approach. Then, the resultant spectrally-coarsened graphs are leveraged for the development of much faster algorithms for multilevel spectral graph embedding (clustering) as well as visualization of large data sets. We conducted extensive experiments using a variety of large graphs and datasets and obtained very promising results. For instance, we are able to coarsen the "coPapersCiteseer" graph with 0.43 million nodes and 16 million edges into a much smaller graph with only 13K (32X fewer) nodes and 17K (950X fewer) edges in about 16 seconds; the spectrally-coarsened graphs allow us to achieve up to 1,100X speedup for multilevel spectral graph embedding (clustering) and up to 60X speedup for t-SNE visualization of large data sets.
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- Award ID(s):
- 2021309
- PAR ID:
- 10252376
- Date Published:
- Journal Name:
- ACM International Conference on Web Search and Data Mining
- Page Range / eLocation ID:
- 869 to 877
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1145/3437963.3441767
