Hierarchical Clustering of Asymmetric Networks

TL;DR

This paper develops hierarchical clustering methods for directed networks with asymmetric dissimilarities, introducing axioms and bounds that define and constrain admissible clustering approaches.

Contribution

It introduces the concept of admissible hierarchical clustering methods for asymmetric networks and characterizes bounds and unique solutions based on modified axioms.

Findings

Reciprocal and nonreciprocal clustering provide bounds for admissible methods.

Modifying axioms leads to a unique admissible clustering method.

The framework applies to directed networks with asymmetric relationships.

Abstract

This paper considers networks where relationships between nodes are represented by directed dissimilarities. The goal is to study methods that, based on the dissimilarity structure, output hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter. Our construction of hierarchical clustering methods is built around the concept of admissible methods, which are those that abide by the axioms of value - nodes in a network with two nodes are clustered together at the maximum of the two dissimilarities between them - and transformation - when dissimilarities are reduced, the network may become more clustered but not less. Two particular methods, termed reciprocal and nonreciprocal clustering, are shown to provide upper and lower bounds in the space of admissible methods. Furthermore, alternative clustering methodologies and axioms are considered. In particular, modifying the axiom of value such that clustering in two-node networks occurs at the minimum of the two dissimilarities entails the existence of a unique admissible clustering method.