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Many complex systems involve direct interactions among more than two entities and can be represented by hypergraphs, in which hyperedges encode higher-order interactions among an arbitrary number of nodes. To analyze structures and dynamics of given hypergraphs, a solid practice is to compare them with those for randomized hypergraphs that preserve some specific properties of the original hypergraphs. In the present study, we propose a family of such reference models for hypergraphs, called the hyper <inline-formula><tex-math notation="LaTeX">$dK$</tex-math></inline-formula>-series, by extending the so-called <inline-formula><tex-math notation="LaTeX">$dK$</tex-math></inline-formula>-series for dyadic networks to the case of hypergraphs. The hyper <inline-formula><tex-math notation="LaTeX">$dK$</tex-math></inline-formula>-series preserves up to the individual node&#x0027;s degree, node&#x0027;s degree correlation, node&#x0027;s redundancy coefficient, and/or the hyperedge&#x0027;s size depending on the parameter values. Furthermore, we numerically find that higher-order hyper <inline-formula><tex-math notation="LaTeX">$dK$</tex-math></inline-formula>-series more accurately preserves the shortest path length and degree distribution of the one-mode projection of the original hypergraph, which the method does not intend to preserve. We also apply the hyper <inline-formula><tex-math notation="LaTeX">$dK$</tex-math></inline-formula>-series to numerical simulations of epidemic spreading and evolutionary game dynamics on empirical social hypergraphs. We find that the hyperedge&#x0027;s size affects these dynamics more than any of the node&#x0027;s properties and that the node&#x0027;s degree correlation and redundancy in the empirical hypergraphs promote cooperation.

Randomizing Hypergraphs Preserving Degree Correlation and Local Clustering