Variational perspective on local graph clustering
Name
1602.01886.pdf
Description
Accepted version
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562.1 KB
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Author(s)
Fountoulakis, Kimon
Roosta-Khorasani, Farbod
Shun, Julian
Cheng, Xiang
Mahoney, Michael W
Date Issued
2019
Journal
Mathematical Programming
Publisher
Springer Nature
Version
Author's final manuscript
Abstract
© 2017, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society. Modern graph clustering applications require the analysis of large graphs and this can be computationally expensive. In this regard, local spectral graph clustering methods aim to identify well-connected clusters around a given “seed set” of reference nodes without accessing the entire graph. The celebrated Approximate Personalized PageRank (APPR) algorithm in the seminal paper by Andersen et al. (in: FOCS ’06 proceedings of the 47th annual IEEE symposium on foundations of computer science, pp 475–486, 2006) is one such method. APPR was introduced and motivated purely from an algorithmic perspective. In other words, there is no a priori notion of objective function/optimality conditions that characterizes the steps taken by APPR. Here, we derive a novel variational formulation which makes explicit the actual optimization problem solved by APPR. In doing so, we draw connections between the local spectral algorithm of Andersen et al. (2006) and an iterative shrinkage-thresholding algorithm (ISTA). In particular, we show that, appropriately initialized ISTA applied to our variational formulation can recover the sought-after local cluster in a time that only depends on the number of non-zeros of the optimal solution instead of the entire graph. In the process, we show that an optimization algorithm which apparently requires accessing the entire graph, can be made to behave in a completely local manner by accessing only a small number of nodes. This viewpoint builds a bridge across two seemingly disjoint fields of graph processing and numerical optimization, and it allows one to leverage well-studied, numerically robust, and efficient optimization algorithms for processing today’s large graphs.
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/S10107-017-1214-8
