Resolution-limit-free and local Non-negative Matrix Factorization quality functions for graph clustering

TL;DR

This paper explores resolution-limit-free and local properties of Non-negative Matrix Factorization (NMF) for graph clustering, proposing new quality functions that overcome limitations of existing methods in both hard and soft clustering scenarios.

Contribution

It introduces a novel class of local probabilistic NMF quality functions for soft graph clustering and analyzes conditions for resolution-limit-free properties in NMF-based clustering.

Findings

Symmetric NMF is resolution-limit-free with hardness constraints.

Local NMF quality functions are proposed for soft clustering.

Resolution-limit-free property is too strong for soft clustering.

Abstract

Many graph clustering quality functions suffer from a resolution limit, the inability to find small clusters in large graphs. So called resolution-limit-free quality functions do not have this limit. This property was previously introduced for hard clustering, that is, graph partitioning. We investigate the resolution-limit-free property in the context of Non-negative Matrix Factorization (NMF) for hard and soft graph clustering. To use NMF in the hard clustering setting, a common approach is to assign each node to its highest membership cluster. We show that in this case symmetric NMF is not resolution-limit-free, but that it becomes so when hardness constraints are used as part of the optimization. The resulting function is strongly linked to the Constant Potts Model. In soft clustering, nodes can belong to more than one cluster, with varying degrees of membership. In this setting resolution-limit-free turns out to be too strong a property. Therefore we introduce locality, which roughly states that changing one part of the graph does not affect the clustering of other parts of the graph. We argue that this is a desirable property, provide conditions under which NMF quality functions are local, and propose a novel class of local probabilistic NMF quality functions for soft graph clustering.