Message passing method for social contagion in hypergraphs

The emergence of hypergraphs has solved the problem that the interactions between nodes are insufficient to describe the complex relationships among multiple individuals. In this paper, we model social contagion with the reinforcement effect on hypergraphs, where hyperedges disseminate information to nodes, and nodes upload information to hyperedges. In order to reduce the complexity of high-order interactions on the propagation, hypergraphs are mapped to factor graphs, where hyperedges are encoded to factor nodes, and the connection between a node and a factor node indicates that the node is located in the hyperedge. Taking into account the heterogeneity of nodes and hyperedges, we establish the message passing evolution equations about each node based on the factor graph. Finally, we carry out numerical simulations by iterating the message passing equations. We find that the probability of the adopted state decreases before the outbreak of social contagion, and the final adopting scale suddenly increases as the transmission rates increase, which are caused by the combined action of high-order interactions and the social reinforcement effect. Significantly, the final adopting scale presents a step-like variation when the adopting threshold of hyperedges changes.