Reducibility of higher-order to pairwise interactions: Social impact models on hypergraphs

We show that a general class of social impact models with higher-order interactions on hypergraphs can be exactly reduced to an equivalent model with pairwise interactions on a weighted projected network. This reduction is made by a mapping that preserves the microscopic probabilities of changing the state of the nodes. As a particular case, we introduce hypergraph-voter models, for which we compute the weights of the projected network both analytically and numerically across several hypergraph ensembles, and we characterize their ordering dynamics through simulations of both higher-order and reduced dynamics. For a linear social impact function (hypergraph-linear voter model) the weights of the projected network are static, allowing us to develop a pair approximation that describes with accuracy the time evolution of macroscopic observables, which turn out to be independent of those weights. The macroscopic dynamics is thus equivalent to that of the standard voter model on the unweighted projected network. For a power-law social impact function (hypergraph-nonlinear voter model) the weights of the projected network depend on the instantaneous system configuration. Nevertheless, the nonlinear voter model on the unweighted projected network still reproduces the main macroscopic trends for well connected hypergraphs.