Hierarchical clustering methods and algorithms for asymmetric networks

Three different families of hierarchical clustering methods satisfying the axioms of value - in a network with two nodes the nodes cluster together at resolutions at which both can influence each other - and transformation - when we reduce some pairwise dissimilarities and increase none, the resolutions at which nodes cluster together may decrease but not increase - are introduced. The grafting family exchanges branches between dendrograms generated by different admissible methods. The convex combination family combines admissible methods using a convex operation in the space of dendrograms. The semi-reciprocal family is related to the reciprocal and nonreciprocal clustering methods introduced in [1]. 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.