Heading in the right direction? Using head moves to traverse phylogenetic network space Remie Janssen Vol. 25, no. 1, pp. 263-310, 2021. Regular paper. Abstract Head moves are a type of rearrangement moves for phylogenetic networks. They have primarily been studied as part of other 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 into 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 small in general, 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. Submitted: February 2019. Reviewed: March 2020. Revised: March 2020. Reviewed: July 2020. Revised: August 2020. Accepted: March 2021. Final: March 2021. Published: April 2021. Communicated by Fabio Vandin article (PDF) BibTeX