Incompatibility boundaries for properties of community partitions

TL;DR

This paper demonstrates that certain desirable properties of community partition quality functions are fundamentally incompatible, but relaxing these properties can lead to viable quality functions, highlighting the complexity of defining optimal community partitions.

Contribution

It generalizes Kleinberg's impossibility result by considering weaker property sets and introduces relaxed properties with example quality functions.

Findings

Incompatibility of certain community partition properties is proven.

Relaxed properties allow for the construction of compatible quality functions.

Experimental results show mixed performance of these functions compared to established methods.

Abstract

We prove the incompatibility of certain desirable properties of community partition quality functions. Our results generalize the impossibility result of [Kleinberg 2003] by considering sets of weaker properties. In particular, we use an alternative notion to solve the central issue of the consistency property. (The latter means that modifying the graph in a way consistent with a partition should not have counterintuitive effects). Our results clearly show that community partition methods should not be expected to perfectly satisfy all ideally desired properties. We then proceed to show that this incompatibility no longer holds when slightly relaxed versions of the properties are considered, and we provide in fact examples of simple quality functions satisfying these relaxed properties. An experimental study of these quality functions shows a behavior comparable to established methods in some situations, but more debatable results in others. This suggests that defining a notion of good partition in communities probably requires imposing additional properties.