Tree-average distances on certain phylogenetic networks have their weights uniquely determined

This paper proposes a tree-average distance for networks more general than trees. The notion requires a weight on each arc measuring the genetic change along the arc. For each displayed tree the distance between two leaves is the sum of the weights along the path joining them. At a hybrid vertex, each character is inherited from one of its parents. We will assume that for each hybrid there is a probability that the inheritance of a character is from a specified parent. Assume that the inheritance events at different hybrids are independent. Then for each displayed tree there will be a probability that the inheritance of a given character follows the tree; this probability may be interpreted as the probability of the tree. The tree-average distance between the leaves is defined to be the expected value of their distance in the displayed trees.For a class of rooted networks that includes rooted trees, it is shown that the weights and the probabilities at each hybrid vertex can be calculated given the network and the tree-average distances between the leaves. Hence these weights and probabilities are uniquely determined. The hypotheses on the networks include that hybrid vertices have indegree exactly 2 and that vertices that are not leaves have a tree-child.In phylogeny, the evolution of a collection of species is modelled via a directed graph in which the vertices are species and the arcs indicate direct descent, usually with modification as mutations accumulate. The leaves typically correspond to extant species, while internal vertices typically correspond to presumed ancestors. It has been common to assume that the directed graphs are trees, but more recently more general networks have also been studied so as to include the possibility of hybridization of species or lateral gene transfer. General frameworks for phylogenetic networks are discussed in [1], [2], [3], and [4]. See also the recent book [5].There are many methods to reconstruct phylogenetic trees from in