  Activities such as the movement of passengers and goods, the transfer of
physical or digital assets, web navigation and even successive passes in
football, result in timestamped paths through a physical or virtual network.
The need to analyse such paths has produced a new modelling paradigm in the
form of higher-order networks which are able to capture temporal and
topological characteristics of sequential data. This has been complemented by
sequence mining approaches, a key example being sequential motifs measuring the
prevalence of recurrent subsequences. Previous work on higher-order networks
has focused on how to identify the optimal order for a path dataset, where the
order can be thought of as the number of steps of memory encoded in the model.
In this paper, we build on these approaches to consider which orders are
necessary to reproduce different path characteristics, from path lengths to
counts of sequential motifs, viewing paths generated from different
higher-order models as null models which capture features of the data up to a
certain order, and randomised otherwise. Furthermore, we provide an important
extension to motif counting, whereby cases with self-loops, starting nodes, and
ending nodes of paths are taken into consideration. Conducting a thorough
analysis using path lengths and sequential motifs on a diverse range of path
datasets, we show that our approach can shed light on precisely where models of
different order overperform or underperform, and what this may imply about the
original path data.
