Hypernetwork science via high-order hypergraph walks

Abstract

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.

Hypergraph models of biological networks to identify genes critical to pathogenic viral response

“…While many hypergraph topological measures are available, either as generalizations of graph measures to account for multi-way interactions or as native hypergraph-only measures, our focus in this paper is applying generalizations of graph centrality measures to hypergraphs built from transcriptomics data to identify important genes. In order to define these hypergraph centrality measures we must first introduce the notions of a hypergraph walk and distance [ 36 ]. Given two hyperedges , an s -walk between e and f is a sequence of hyperedges such that , , and for all .…”

“…Continuing to follow Aksoy et al [ 36 ], for a fixed , we define the s -distance between two edges as the shortest length of the possibly many s -walks between them. If there is no s -walk between two edges then the s -distance is infinite.…”

“…In order to take into account multiple s values simultaneously in our analysis we developed the average s -betweenness centrality and average harmonic s -closeness centrality , averaging the centrality values defined in [ 36 ] across a range of s values. Our average s -centralities are defined as Computing average s -centralities for each hyperedge provides a ranked list of hyperedges from most central (high value) to least central.…”

Persistent spectral graph

“…Thus it provides a powerful tool to encode highdimensional datasets into various topological and geometric features in a coherent fashion. 2…”

“…However, traditional graphs only represent the pairwise relationships between entries. Therefore, hypergraphs, a generalization of graphs that describe the multi-way relationships of mathematical structures have been developed to capture the high-level complexity of data [2,6]. Mathematically, graphs and hypergraphs are intrinsically related to the simplicial complexes, which have broader use in computational topology.…”