Complex hypernetworks are generalizations of complex networks in which the edges, called hyperedges, can relate more than two nodes. In particular, scale-free hypernetworks can be obtained as evolving networks, using a generalization of a “preferential attachment” algorithm. Hypernetworks occur in natural way in various models in social and economic sciences, where the nodes represent the agents and hyperedges represent relationships or interactions involving more than two agents which lead to the appearance of new, so-called emergent phenomena (e.g., groups of authors who published papers together, three-way relationships between resources, tags, and users in the folksonomy, etc.). So far little has been known about critical phenomena in systems with the structure of complex hypernetworks. In this work, p-spin models on complex scale-free hypernetworks are investigated by means of Monte Carlo simulations, with the spins located in the nodes and the hyperedges connecting p nodes corresponding to ferromagnetic p-spin exchange interactions. In particular, systems without time-reversal symmetry, with p = 3, 5, 7… are considered. Studying phase transitions in such systems is a natural extension of the previous investigations of the ferromagnetic transition in the Ising model on scale-free networks. The models considered in this work differ from the mean-field p-spin models which have often been applied to study spin-glass transition mainly by the presence of the topological structure of interactions. In the systems under study the ground state can be highly degenerate: many spin-glass-like states, with apparently random orientations of spins, can have energy equal to that of the ferromagnetic ordered state. The degeneracy of the ground state is determined by the hypernetwork topology, which in turn depends on the details of the “preferential attachment” algorithm used to build the hypernetwork. If in the thermodynamic limit the degeneracy is finite, evidence for the phase transition is found, e.g., the cusp in the magnetic susceptibility, and below the critical temperature the ferromagnetic state is a possible stable phase. In contrast, if there are infinitely many degenerate ground states, neither ferromagnetic nor spin-glass transition is observed.

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