Randomizing Hypergraphs Preserving Degree Correlation and Local Clustering

Abstract

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 dK -series, by extending the socalled dK -series for dyadic networks to the case of hypergraphs. The hyper dK -series preserves up to the individual node's degree, node's degree correlation, node's redundancy coefficient, and/or the hyperedge's size depending on the parameter values. Furthermore, we numerically find that higher-order hyper dKseries 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 dK -series to numerical simulations of epidemic spreading and evolutionary game dynamics on empirical social hypergraphs. We find that the hyperedge's size affects these dynamics more than any of the node's properties and that the node's degree correlation and redundancy in the empirical hypergraphs promote cooperation.

Fixation dynamics on hypergraphs

“…However, the complete 3-uniform hypergraph with N > 4 is neither amplifier nor suppressor of selection. This is because Eq (52) implies that x 2 = 2 3−N when r = 1, which is different from the result for the Moran process, i.e., x 2 = 2/N, when N > 4.…”

“…Therefore, we compare x 2 , instead of x 1 , as a function of r with x 2 for the Moran process, to examine whether a given hypergraph is an amplifier of selection, suppressor of selection, equivalent to the Moran process, or neither. We compare x 2 calculated from Eq (52) with that for the Moran process at four values of N in Fig 4 . The figure indicates that the complete 3-uniform hypergraph with 4 nodes is a suppressor of selection. However, the complete 3-uniform hypergraph with N > 4 is neither amplifier nor suppressor of selection.…”

Fixation dynamics on hypergraphs

“…We obtained the randomized hypergraph for each empirical hypergraph by randomly shuffling the hyperedges of the original hypergraph. In the random shuffling, we preserved the degree of each node and the size of each hyperedge [47,48]. We show the fixation probability for the randomized hypergraphs by the green circles in Fig.…”

Higher-order rich-club phenomenon in collaborative research grants

“…For dyadic networks, a standard choice of the reference model is the configuration model, which randomizes the edges of the original network while preserving the degree of each node 61 . Here we use a counterpart of the configuration model for bipartite networks in which we randomize the edges of the original bipartite network while preserving the degree of each institution and each collaborative grant 65,66 . We compute φ rand (k) as the rich-club coefficient averaged over 10,000 randomized bipartite networks.…”