We study analytically and numerically the statics and the off-equilibrium dynamics of spin-models over finitely connected random graphs. The analysis of the ground state phase diagram allows us to identify a threshold value for the connectivity beyond which the loop structure of the graph becomes thermodynamically relevant. In hypergraphs, such a percolation of order transition appears above the percolation transition, with a first order mechanism. Extensive Glauber dynamics simulations allow us to show that such a loop structure is responsible for the onset of dynamical features of a local character, like dynamical heterogeneities and a spontaneous time scale separation, consistently with previous (experimental and numerical) studies of structural glasses and spin glasses in their approach to the low temperature phase.

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