Variational perspective on local graph clustering

• 成果类型: Article
• 署名作者: Fountoulakis, Kimon; Roosta-Khorasani, Farbod; Shun, Julian; Cheng, Xiang; Mahoney, Michael W.
• 署名单位: University of California System; University of California Berkeley; University of California System; University of California Berkeley
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-017-1214-8
• 发表日期: 2019
• 页码: 553-573

摘要

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.