Cooperative and competitive hyperorder interactions nonlinearly reshape the collective dynamics in complex networks.

Abstract

Higher-order interactions have recently emerged as key determinants of collective dynamics in complex networks, driving phenomena such as synchronization and multistability. However, how different higher-order groups interact and jointly shape global behavior remains poorly understood. Here, we introduce the model of hyperorder interactions (HpOI), a framework that captures competitive and cooperative couplings between higher-order groups. Using numerical simulations of real-world networks (neural, social, and natural systems) and theoretical analysis based on the Ott-Antonsen method, we reveal a nonmonotonic effect of HpOI on synchronization stability. Cooperative HpOI consistently enlarges the synchronization region. By contrast, the effect of competitive HpOI depends on the overlap among higher-order groups: under low overlap, strong competition can also enhance synchronization stability, whereas under high overlap, it suppresses global synchronization. This work incorporates coupling between higher-order groups into higher-order networks and provides insights into structure-function coupling within real-world systems.