Phylogenetics is concerned with the construction and analysis of evolutionary or phylogenetic trees and networks to understand the evolution of species, populations and individuals . Neighbor-Net is a phylogenetic analysis and data representation method introduced in . It is loosely based on the popular Neighbor-Joining (NJ) method of Saitou and Nei , but with one fundamental difference: whereas NJ constructs phylogenetic trees, Neighbor-Net constructs phylogenetic networks. The method is widely used, in areas such as virology , bacteriology , plant evolution  and even linguistics .
Evolutionary processes such as hybridization between species, lateral transfer of genes, recombination within a population, and convergent evolution can all lead to evolutionary histories that are distinctly non tree-like. Moreover, even when the underlying evolution is tree-like, the presence of conflicting or ambiguous signal can make a single tree representation inappropriate. In these situations, phylogenetic network methods can be particularly useful (see e.g. ).
Phylogenetic networks are a generalization of phylogenetic trees (see Figure 1 for a typical example of a phylogenetic network). In case there are many conflicting phylogenetic signals supported by the data, Neighbor-Net can represent this conflict graphically. In particular a single network can represent several trees simultaneously, indicate whether or not the data is substantially tree-like, and give evidence for possible reticulation or hybridization events. Evolutionary hypotheses suggested by the network can be tested directly using more detailed phylogenetic analyses and specialized biochemical methods (e.g. DNA fingerprinting or chromosome painting).
For any network construction method, it is vital that the network does not depict more conflict than is found in the data and that, if there are conflicting signals, then these should be represented by the network. At the same time, when the data is fitted well by a tree, the method should return a network that is close to being a tree. This is essential not just to avoid false inferences, but for the application of networks in statistical tests of the extent to which the data is tree-like .
In this paper we provide a proof that these properties all hold for Neighbor-Net. Formally, we prove that if the input to NeighborNet is a circular distance function (distance matrix) , then the method returns a network that exactly represents the distance. Circular distance functions are more general than additive (patristic) distances on trees and, thus, as a corollary, if Neighbor-Net is given an additive distance it will return the corresponding tree. In this sense, Neighbor-Net is a statistically consistent method.
The paper is structured as follows: In Section 2 we introduce some basic notation, and in Section 3 we review the Neighbor-Net algorithm. In Section 4 we prove that Neighbor-Net is consistent (Theorem 4.1).