Supporting Information: a mathematical theory of explicit phylogenetic networks with edge lengths.

This document provides an introduction to the mathematical theory of explicit phylogenetic networks with edge lengths, leading in particular to the proofs of Propositions 1 and 2, which are necessary for the proof of Theorem 1, part (ii). In the last section, we consider networks with inheritance probabilities and their relevance for likelihood-based reconstruction. (PDF)本文系统介绍了带边长度的显式系统发育网络（phylogenetic networks）的数学理论，并重点给出了命题1与命题2的证明，二者是定理1第(ii)部分证明的必要前提。最后一节探讨了带有遗传概率的系统发育网络，及其在基于似然的重建方法中的应用意义。(PDF)