Why are splits graphs still called phylogenetic networks?

This is an issue that has long concerned me, and which I think causes a lot of confusion among biologists. A phylogenetic tree is usually a clear concept — to a biologist, it is a diagram that displays a hypothesis of evolutionary history. The expectation, then, is that a phylogenetic network does the same thing for reticulate evolutionary histories. However, this is not true of splits graphs; and so there is potential confusion.

Mathematically, of course, a phylogenetic tree is a directed acyclic line graph. It is usually constructed, in practice, by first producing an undirected graph based on some pattern-analysis procedure, and then nominating one of the nodes or edges as the root (say, by specifying an outgroup). So, the mathematics is not really connected to the biological interpretation. To a mathematician, the tree is a set of nodes connected by directed edges, and the nodes could represent anything at all, as could the edges. It is the biologist who artificially imposes the idea that the nodes represent real historical organisms connected by the flow of evolution — ancestors connected to descendants by evolutionary events.

A phylogenetic network should logically be a generalization of this idea of a phylogenetic tree, adding the possibility of evolutionary relationships due to gene flow, in addition to the ancestor-descendant relationships. This can be done, but it is only partly done by splits graphs.

That is, a splits graph generalizes the idea of an undirected line graph (an unrooted tree), but not a directed acyclic graph (a rooted tree). It follows the same logic of using a pattern-analysis procedure to produce an undirected graph, although the graph can have reticulations, and thus is a network rather than necessarily being a bifurcating tree. However, it is not straightforward to specify a root in a way that will turn this into an acyclic graph. So, in general it does not represent a phylogeny.

Indeed, splits graphs are simply one form of multivariate pattern analysis, along with clustering and ordination techniques, which are familiar as data-display methods in phenetics (see Morrison D.A. 2014. Phylogenetic networks — a new form of multivariate data summary for data mining and exploratory data analysis. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery 4: 296-312). In this sense, it makes no difference whatsoever what the data represent — they can be data used for phylogenetics, or they could be any other form of multivariate data. Indeed, this point is illustrated in many of the posts in this blog, which can be accessed in the Analyses page.

So, unlike unrooted trees, unrooted splits graphs are not a route to producing a phylogenetic diagram. Mind you, they are a very useful form of multivariate data analysis in their own right, and I value them highly as a form of exploratory data analysis. But that doesn't make them phylogenetic networks in the biological sense.

So, isn't it about time we stopped calling splits graphs "phylogenetic networks"? They aren't, to a biologist, so why call them that?