  We propose high-order hypergraph walks as a framework to generalize
graph-based network science techniques to hypergraphs. Edge incidence in
hypergraphs is quantitative, yielding hypergraph walks with both length and
width. Graph methods which then generalize to hypergraphs include connected
component analyses, graph distance-based metrics such as closeness centrality,
and motif-based measures such as clustering coefficients. We apply high-order
analogs of these methods to real world hypernetworks, and show they reveal
nuanced and interpretable structure that cannot be detected by graph-based
methods. Lastly, we apply three generative models to the data and find that
basic hypergraph properties, such as density and degree distributions, do not
necessarily control these new structural measurements. Our work demonstrates
how analyses of hypergraph-structured data are richer when utilizing tools
tailored to capture hypergraph-native phenomena, and suggests one possible
avenue towards that end.
