Explosive neural networks via higher-order interactions in curved statistical manifolds

Miguel Aguilera, Pablo A. Morales, Fernando E. Rosas, Hideaki Shimazaki

Abstract: Higher-order interactions underlie complex phenomena in systems such as biological and artificial neural networks, but their study is challenging due to the scarcity of tractable models. By leveraging a generalisation of the maximum entropy principle, here we introduce curved neural networks as a class of prototypical models with a limited number of parameters that are particularly well-suited for studying higher-order phenomena. Through exact mean-field descriptions, we show that these curved neural networks implement a self-regulating annealing process that can accelerate memory retrieval, leading to explosive order-disorder phase transitions with multi-stability and hysteresis effects. Moreover, by analytically exploring their memory-retrieval capacity using the replica trick near ferromagnetic and spin-glass phase boundaries, we demonstrate that these networks can enhance memory capacity and robustness of retrieval over classical associative-memory networks. Overall, the proposed framework provides parsimonious models amenable to analytical study, revealing novel higher-order phenomena in complex networks.