Non-dyadic higher-order interactions affect collective behavior in various networked dynamical systems. Here, we discuss the properties of a novel Ising model with higher-order interactions and characterize its phase transitions between the ordered and the disordered phase. By a mean-field treatment, we show that the transition is continuous when only three-body interactions are considered, but becomes abrupt when interactions of higher orders are introduced. Using a Georges-Yedidia expansion to go beyond a naïve mean-field approximation, we reveal a quantitative shift in the critical point of the phase transition, which does not affect the universality class of the model. Finally, we compare our results with traditional $p$-spin models with many-body interactions. Our work unveils new collective phenomena on complex interacting systems, revealing the importance of investigating higher-order systems beyond three-body interactions.

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• Collective behavior in networks
• Phase transitions
• Networks & random structures
• Ising model