# SIS Epidemic Propagation on Scale-Free Hypernetwork
## 1 Introduction
Studies on static topology indicators include subgraph centrality and clustering coefficient [14], and other indicators include hyperdegree of node, degree of hyperedge, hyperedge overlap, etc. In terms of dynamic evolution model construction of hypernetwork, [PERSON] [15] proposed a hypernetwork evolution model based on growth and preferential attachment mechanism, in which \"one old node and several new nodes generate a new hyperedge\". In like manner, [PERSON] [16] proposed that \"several old and new nodes generate a new hyperedge\" of the hypernetwork evolution model. On the basis of the above two evolution models, [PERSON] [17] proposed a unified evolution model of hypernetwork where \"several old nodes and several new nodes generate several new hyperedges\" and in this paper, we call it \"universal scale-free hypernetwork\". In the hypernetworks obtained by the above three construction models, the hyperdegree distributions of their nodes all follow the power-law form and have scale-free characteristics. In the research of complex networks, the simulation of propagation process based on pairwise interaction has been widely and deeply studied. However, in many scenarios, communication in the real system may be realized through multiple interactions, while higher-order propagation refers to the propagation process that requires multiple interactions and takes place in a higher-order structure. The research on higher-order propagation can explain the propagation phenomenon from a new perspective and discover the dynamics properties missing from the perspective of pairwise interaction. Propagation models cover many aspects, from different types to richer substrates underlying the process itself. Nevertheless, as recently argued in [18], real data are revealing that pairwise relationships--the fundamental interaction units of networks--do not capture complex dependencies. Therefore, higher-order interactions are ubiquitous, and understanding their properties and impacts is of paramount importance. Combinatorial higher-order models [18] offer a way to describe these systems as they overcome the limitations of lower-order network models. In a first attempt, [PERSON] [19] proposed an SIS disease spreading model in a hypergraph. Next, [PERSON] [20] presented a higher-order model of social contagion defined on simplicial complexes and provided approximate solutions for complexes of order three. [PERSON] [21] studied the SIS propagation problem on a scale-free uniform hypergraph, explored the influence of hub nodes on the propagation, and used analytical and numerical methods to determine the critical exponent for the occurrence of infectious disease transitions. Furthermore, [PERSON] [22] investigated the propagation process on hypergraphs and generalized it to several mean-field and heterogeneous mean-field approaches, where continuous and discontinuous phase transitions are observed from higher-order propagation, and critical mass effects emerge. [PERSON] [23] analyzed a SIS model of infectious disease spread on hypergraphs and found that due to the newly introduced higher-order interaction terms, a new dynamical behavior domain emerged: Disease-free equilibrium and endemic equilibrium coexist, and are locally asymptotically stable. Importantly, [PERSON] [24] leveraged a hypergraph model to certify the heterogeneity of environments and the heterogeneity of individual participation in these environments and found that under nonlinear infection kernels, conventional epidemic wisdom breaks down with the emergence of discontinuous transitions, super-exponential spread, and hysteresis. All of the above models presented new phenomenological patterns associated with the critical properties of the dynamics. However, those proposed models are still very constrained, both structurally and dynamically. Here, we extend their model both structurally and dynamically. Structurally, we use hypergraphs, which generalize the concept of graphs, by allowing an edge to have an arbitrary number of nodes and generated a universal hypernetwork.