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

Miguel Aguilera (), Pablo A. Morales, Fernando E. Rosas and Hideaki Shimazaki
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Miguel Aguilera: BCAM – Basque Center for Applied Mathematics
Pablo A. Morales: Araya Inc.
Fernando E. Rosas: University of Sussex
Hideaki Shimazaki: Kyoto University

Nature Communications, 2025, vol. 16, issue 1, 1-10

Abstract: 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, we introduce curved neural networks as a class of 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, 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 higher-order phenomena in complex networks.

Date: 2025
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DOI: 10.1038/s41467-025-61475-w

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