Robust Hierarchical Clustering for Directed Networks: An Axiomatic Approach

TL;DR

This paper characterizes robust hierarchical clustering methods for directed networks using an axiomatic approach, introducing properties like scale preservation, stability, and excisiveness, and providing a generative model for these methods.

Contribution

It provides a complete axiomatic characterization of robust hierarchical clustering methods for directed networks, introducing the concept of representability and analyzing their properties.

Findings

Characterization of all robust clustering methods using the proposed axioms.

Introduction of the concept of representability for clustering methods.

Application of the methods to real data demonstrating practical utility.

Abstract

We provide a complete taxonomic characterization of robust hierarchical clustering methods for directed networks following an axiomatic approach. We begin by introducing three practical properties associated with the notion of robustness in hierarchical clustering: linear scale preservation, stability, and excisiveness. Linear scale preservation enforces imperviousness to change in units of measure whereas stability ensures that a bounded perturbation in the input network entails a bounded perturbation in the clustering output. Excisiveness refers to the local consistency of the clustering outcome. Algorithmically, excisiveness implies that we can reduce computational complexity by only clustering a subset of our data while theoretically guaranteeing that the same hierarchical outcome would be observed when clustering the whole dataset. In parallel to these three properties, we introduce the concept of representability, a generative model for describing clustering methods through the specification of their action on a collection of networks. Our main result is to leverage this generative model to give a precise characterization of all robust -- i.e., excisive, linear scale preserving, and stable -- hierarchical clustering methods for directed networks. We also address the implementation of our methods and describe an application to real data.