Despite this preliminary evidence, however, a rigorousand general proof of this conjecture is still lacking.Approximate approaches based on linearization arounda fixed point of ordinary differential equations linkthe stability of hypergraphs dynamics to their graphprojections [ 51 – 53 ], suggesting general conditions forstability associated to the different orders of the inter-actions. Mean-field treatment allows for an analyticalsolution for diffusion and spreading processes on arbitrarystructures, separating stability conditions into structuraland dynamical terms [54 , 55 ]. A general argument basedon bifurcation theory shows that variations on pairwisemodels, such as adding higher-order interactions, canlead to a change of critical behaviour from a continuousto a discontinuous transition for a wide class of models