  Contagion processes have been proven to fundamentally depend on the
structural properties of the interaction networks conveying them. Many real
networked systems are characterized by clustered substructures representing
either collections of all-to-all pair-wise interactions (cliques) and/or group
interactions, involving many of their members at once. In this work, focusing
on interaction structures represented as simplicial complexes, we present a
discrete-time microscopic model of complex contagion for a
susceptible-infected-susceptible dynamics. Introducing a particular edge clique
cover and a heuristic to find it, the model accounts for the higher-order
dynamical correlations among the members of the substructures
(cliques/simplices). The analytical computation of the critical point reveals
that higher-order correlations are responsible for its dependence on the
higher-order couplings. While such dependence eludes any mean-field model, the
possibility of a bi-stable region is extended to structured populations.
