# Sequential motifs in observed walks

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#### Abstract

The structure of complex networks can be characterized by counting and analysing network *motifs*. Motifs are small graph structures that occur repeatedly in a network, such as triangles or chains. Recent work has generalized motifs to temporal and dynamic network data. However, existing techniques do not generalize to *sequential* or *trajectory* data, which represent entities moving through the nodes of a network, such as passengers moving through transportation networks. The unit of observation in these data is fundamentally different since we analyse observations of trajectories (e.g. a trip from airport A to airport C through airport B), rather than independent observations of edges or snapshots of graphs over time. In this work, we define *sequential motifs* in trajectory data, which are small, directed and sequence-ordered graphs corresponding to patterns in observed sequences. We draw a connection between the counting and analysis of sequential motifs and Higher-Order Network (HON) models. We show that by mapping edges of a HON, specifically a kkth-order DeBruijn graph, to sequential motifs, we can count and evaluate their importance in observed data. We test our methodology with two datasets: (1) passengers navigating an airport network and (2) people navigating the Wikipedia article network. We find that the most prevalent and important sequential motifs correspond to intuitive patterns of traversal in the real systems and show empirically that the heterogeneity of edge weights in an observed higher-order DeBruijn graph has implications for the distributions of sequential motifs we expect to see across our null models.