The phase polarization induced by homogeneous structure of evolving random hypergraph

Hypergraphs are an extremely useful tools that have been used extensively in recent years to study complex systems with nonlinearity, time delays, cooperation, and other complex dynamics. The generation of random hypergraphs for the investigation of network structures is an essential direction in hypergraph research. In network science, the Preferential Attachment (PA) paradigm is central for generating random hypergraph networks. However, previous studies have predominantly focused on PA mechanisms based on cooperative patterns, while ignoring the inclusion of affiliation and the group evolution in the emergence of hyperedges. In this paper, we develop a random hypergraph evolution model based on the consensus threshold mechanism. We investigate the degree distribution and the hyperedge evolution mechanisms numerically and theoretically. Moreover, we propose a hypergraph homogeneity index to quantitatively characterize hypergraph structures. Unlike classical centrality-based metrics, this index effectively captures the degree-distribution patterns within hyperedges and extends the concept of degree-based distance. By introducing the homogeneity index, our results reveal that the structural properties of hypergraphs are significantly influenced by the consensus mechanism. A flatter network topology is implied by a higher threshold; however, it also indicates a more homogeneous structure. Furthermore, based on our proposed model, we also investigate how homogeneity affects the polarization process within different network structures. Heterogeneous networks are less prone to polarization, while homogeneous hypergraphs are more susceptible to polarization phenomena.