Higher-order components in hypergraphs

Most hypergraph studies have been focused on the interaction within a hyperedge but not on the interaction between hyperedges. We introduce s-connectivity in hypergraphs as the connectivity between two hyperedges sharing s common nodes and s-component as the connected component only through s- or higher-order connectivities. Such higher-order components are observed ubiquitously in real-world hypergraphs but lack in their degree-preserved randomized counterparts. To this end, we propose a novel random hypergraph model in which higher-order components substantially exist. The concept of the group consisting of randomly preassigned nodes is introduced in the model. Our hypergraph model evolves by recruiting either a random node (with probability 1-p) or a random group (with probability p) to a random hyperedge until the desired mean degree is reached. We analytically compute the properties of the model hypergraphs, such as the giant-s-component size, supported by numerical simulations. A series of s-component percolation transitions is discovered. Implications of the s-connectivity to the structure and dynamics of hypergraph systems are also investigated. We anticipate that this model could provide a framework for exploring the higher-order architectures in hypergraphs.