A parameterized family of balance indices for phylogenetic networks

Abstract: We introduce a new family of balance indices for phylogenetic networks: the $H_α$ indices, where $α$ is a positive real number. This family includes the $B_2$ index as a special case ($α= 1$) and provides a natural extension of the Sackin index to phylogenetic networks. We show that the $H_α$ indices share many structural properties with the $B_2$ index, most notably a "grafting property" that makes it possible to express the $H_α$ index of a network in terms of the $H_α$ indices of its biconnected components. These properties allow us to identify networks that minimize / maximize $H_α$ for various classes of phylogenetic networks, and to study its distribution for several models of random trees and networks (in particular, Galton-Watson trees and binary Markov branching trees, with a focus on the Yule and PDA models). Finally, we show how local limits can be used to analyze the asymptotic behavior of $H_α$ for large trees and networks, and we obtain general results for the moments of $H_α$ for a broad class of random phylogenetic networks known as blowups of Galton-Watson trees.