Admissible Hierarchical Clustering Methods and Algorithms for Asymmetric Networks

TL;DR

This paper characterizes admissible hierarchical clustering methods for asymmetric networks, introduces new intermediate methods, and provides algorithms for their efficient computation, demonstrated through an application to the U.S. economy network.

Contribution

It defines admissible clustering methods for asymmetric networks, introduces three new families of methods, and develops algorithms for their practical computation.

Findings

Admissible methods are bounded by reciprocal and nonreciprocal clustering.

Algorithms for various clustering methods are derived using matrix operations in dioid algebra.

Application to U.S. economy network illustrates the methods' utility.

Abstract

This paper characterizes hierarchical clustering methods that abide by two previously introduced axioms -- thus, denominated admissible methods -- and proposes tractable algorithms for their implementation. We leverage the fact that, for asymmetric networks, every admissible method must be contained between reciprocal and nonreciprocal clustering, and describe three families of intermediate methods. Grafting methods exchange branches between dendrograms generated by different admissible methods. The convex combination family combines admissible methods through a convex operation in the space of dendrograms, and thirdly, the semi-reciprocal family clusters nodes that are related by strong cyclic influences in the network. Algorithms for the computation of hierarchical clusters generated by reciprocal and nonreciprocal clustering as well as the grafting, convex combination, and semi-reciprocal families are derived using matrix operations in a dioid algebra. Finally, the introduced clustering methods and algorithms are exemplified through their application to a network describing the interrelation between sectors of the United States (U.S.) economy.