Improved Graph Clustering

TL;DR

This paper introduces a convexified maximum likelihood algorithm for graph clustering that significantly improves performance in the stochastic block model and its extensions, handling various complex graph scenarios with near-optimal guarantees.

Contribution

The paper presents a novel convexified maximum likelihood method that outperforms existing algorithms in stochastic block models and extends to complex graph settings with strong theoretical guarantees.

Findings

Outperforms existing methods polynomially in classic stochastic block models.

Achieves near-optimal recovery guarantees, close to spectral lower bounds.

Handles diverse graph models including semi-random, heterogeneous, and partially observed graphs.

Abstract

Graph clustering involves the task of dividing nodes into clusters, so that the edge density is higher within clusters as opposed to across clusters. A natural, classic and popular statistical setting for evaluating solutions to this problem is the stochastic block model, also referred to as the planted partition model. In this paper we present a new algorithm--a convexified version of Maximum Likelihood--for graph clustering. We show that, in the classic stochastic block model setting, it outperforms existing methods by polynomial factors when the cluster size is allowed to have general scalings. In fact, it is within logarithmic factors of known lower bounds for spectral methods, and there is evidence suggesting that no polynomial time algorithm would do significantly better. We then show that this guarantee carries over to a more general extension of the stochastic block model. Our method can handle the settings of semi-random graphs, heterogeneous degree distributions, unequal cluster sizes, unaffiliated nodes, partially observed graphs and planted clique/coloring etc. In particular, our results provide the best exact recovery guarantees to date for the planted partition, planted k-disjoint-cliques and planted noisy coloring models with general cluster sizes; in other settings, we match the best existing results up to logarithmic factors.