Fig. 5: An example of optimal network with directed 2-hyperlink interactions that preserves symmetry.

a An illustration on the direction in directed 2-hyperlink interactions. The tensor \({a}_{i,j,k}^{(2)}\) has its subscript indices separately as source, 1st target, 2nd target nodes as in⁵⁶. Two directed 2-hyperlinks corresponding to \({a}_{1,3,5}^{(2)}=1\), \({a}_{1,5,3}^{(2)}=1\) are shown. b The 7-node network with directed 2-hyperlink interactions. The colored triangles denote various directed 2-hyperlink interactions, as given by Eq. (17). For example, two directed 2-hyperlinks \({a}_{1,3,5}^{(2)}=1\), \({a}_{1,5,3}^{(2)}=1\) in total have the arrows 1 → 3, 1 → 5, 3 → 5, 5 → 3 by the rule defined in panel a for setting the directed 2-hyperlink. This 7-node network supports that the structurally symmetric network can be optimal in synchronization for 2-hyperlink interactions. Differently, the optimal directed networks with pairwise interactions are always structurally asymmetric²⁷.