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