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Recently there has been an increasing interest in studying dynamicalprocesses on networks exhibiting higher-order structures, such as simplicialcomplexes, where the dynamics acts above and beyond dyadic interactions. Usingsimulations or heuristically derived epidemic spreading models it was shownthat new phenomena can emerge, such as bi-stability/multistability. Here, weshow that such new emerging phenomena do not require complex contact patterns,such as community structures, but naturally result from the higher-ordercontagion mechanisms. We show this by deriving an exact higher-order SIS modeland its limiting mean-field equivalent for fully connected simplicialcomplexes. Going beyond previous results, we also give the global bifurcationpicture for networks with 3- and 4-body interactions, with the latter allowingfor two non-trivial stable endemic steady states. Differently from previousapproaches, we are able to study systems featuring interactions of arbitraryorder. In addition, we characterise the contributions from higher-orderinfections to the endemic equilibrium as perturbations of the pairwisebaseline, finding that these diminish as the pairwise rate of infectionincreases. Our approach represents a first step towards a principledunderstanding of higher-order contagion processes beyond triads and opens upfurther directions for analytical investigations.

Insights from exact social contagion dynamics on networks with higher-order structures