Variational Perspective on Local Graph Clustering

TL;DR

This paper introduces a variational formulation for local graph clustering, connecting spectral methods with optimization algorithms, enabling efficient, localized clustering on large graphs by only accessing relevant nodes.

Contribution

It derives a variational formulation of APPR, linking it to ISTA, and demonstrates how optimization algorithms can perform local clustering efficiently.

Findings

The variational formulation explicitly characterizes the optimization problem solved by APPR.

Appropriate initialization of ISTA can recover local clusters efficiently.

Optimization algorithms can be adapted to operate locally, reducing graph access requirements.

Abstract

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. 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 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.