Fig. 6: The optimized network with directed 2-hyperlink interactions are structurally asymmetric in general.

Synchronizability of networks with directed 2-hyperlink interactions is optimized by minimizing Eq. (21) with simulated annealing. The directed 2-hyperlink interactions are deleted, which allows to reach directed networks. a, b Violin distribution of the two asymmetric measures in Eq. (25), Eq. (26) for the optimized directed networks with various size 6, 10, 20, 50, 80, 100, as specified by the color. They show that the optimized network are generally asymmetric, as most of the 30 numerical replicates lead to nonzero asymmetric measures. Each numerical replicate has used the most synchronizable network optimized from 1000 different randomly initialized networks. c The network density as the averaged generalized nodes' degree in Eq. (8).