Relationships Among Phylogenetic Networks

Abstract

The underlying reality of a succession of interbreeding populations is a vastly complicated network N. Since Darwin, species trees have been used as a simplified description of the relationships which summarize the overly complicated network N. Recent evidence of hybridization and lateral gene transfer, however, suggest that there are situations where trees are inadequate. Consequently it is important to determine properties that characterize networks closely related to N and possibly more complicated than trees but lacking the full complexity of N. A connected surjective digraph map (CSD) is a map f from one network N to another network M which either collapses an arc to a single point or takes an arc to an arc, which is surjective, and such that the inverse image of a point is always connected. CSD maps are shown to behave well under composition. If there is such a CSD map, the network M is shown to arise naturally as a quotient structure from N. It is proved that if there is a CSD map from N to M, then there is in a way to lift an undirected version of M into N, possibly with added resolution. A CSD map from N to M puts strong constraints on N; if the map were not connected, there would be minimal constraints. A procedure is defined, given N, to construct a standard successively cluster-distinct network from N. In general, it may be useful to study classes of networks such that, for any N, there exists a CSD map from N to some standard member of that class.