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Networks in nature have complex interactions among agents. One significantphenomenon induced by interactions is synchronization of coupled agents, andthe interactive network topology can be tuned to optimize synchronization. Theprevious studies showed that the optimized conventional network with pairwiseinteractions favors a homogeneous degree distribution of nodes when theinteraction is undirected, and is always structurally asymmetric when theinteraction is directed. However, the optimal control on synchronization fornetworks with prevailing higher-order interactions is less explored. Here, byconsidering the higher-order interactions in hypergraph and the Kuramoto modelwith 2-hyperlink interactions, we find that the network topology with optimizedsynchronizability may have distinct properties. For the undirected interaction,optimized networks with 2-hyperlink interactions by simulated annealing tend tobecome homogeneous in the nodes' generalized degree, consistent with1-hyperlink (pairwise) interactions. We further define the directed hyperlink,and rigorously demonstrate that for the directed interaction, the structuralsymmetry can be preserved in the optimally synchronizable network with2-hyperlink interactions, in contrast to the conclusion for 1-hyperlinkinteractions. The results suggest that controlling the network topology ofhigher-order interactions leads to synchronization phenomena beyond pairwiseinteractions.

Optimizing higher-order network topology for synchronization of coupled phase oscillators