Heading in the right direction? Using head moves to traverse phylogenetic network space

Head moves are a type of rearrangement moves for phylogenetic networks. They have mostly been studied as part of more encompassing types of moves, such as rSPR moves. Here, we study head moves as a type of moves on themselves. We show that the tiers ($k>0$) of phylogenetic network space are connected by local head moves. Then we show tail moves and head moves are closely related: sequences of tail moves can be converted to sequences of head moves and vice versa, changing the length by at most a constant factor. Because the tiers of network space are connected by rSPR moves, this gives a second proof of the connectivity of these tiers. Furthermore, we show that these tiers have small diameter by reproving the connectivity a third time. As the head move neighbourhood is in general small, this makes head moves a good candidate for local search heuristics. Finally we prove that finding the shortest sequence of head moves between two networks is NP-hard.