Performance of higher-order networks in reconstructing sequential paths: from micro to macro scale

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 article, 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 randomize 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.