Phylogenetic networks generalize phylogenetic trees in order to model reticulation events. Although the comparison of phylogenetic trees is well studied, and there are multiple ways to do it in an efficient way, the situation is much different for phylogenetic networks. Some classes of phylogenetic networks, mainly tree-child networks, are known to be classified efficiently by their $\mu$-representation, which essentially counts, for every node, the number of paths to each leaf. In this paper, we introduce the extended $\mu$-representation of networks, where the number of paths to reticulations is also taken into account. This modification allows us to distinguish orchard networks and to define a sound metric on the space of such networks that can, moreover, be computed efficiently. The class of orchard networks, as well as being one of the classes with biological significance (one such network can be interpreted as a tree with extra arcs involving coexisting organisms), is one of the most generic ones (in mathematical terms) for which such a representation can (conjecturally) exist, since a slight relaxation of the definition leads to a problem that is Graph Isomorphism Complete.

翻译：系统发生网络推广了系统发生树，以便对网状进化事件进行建模。尽管系统发生树的比较研究已相当完善，并且存在多种高效比较方法，但系统发生网络的情况却大不相同。已知某些系统发生网络类别（主要是树子网络）可通过其$\mu$-表示法高效分类，该表示法本质上统计每个节点到各叶节点的路径数量。本文引入网络的扩展$\mu$-表示法，其中还考虑了到网状节点的路径数量。这一改进使我们能够区分果园网络，并在此类网络空间上定义一种合理的度量，该度量可被高效计算。果园网络类别既具有生物学意义（此类网络可被解释为带有涉及共存物种的额外弧的树），也是数学上最通用的类别之一，因为对定义的轻微放松将导致图同构完全问题。