  We consider a pair of collectively oscillating networks of dynamical elements
and optimize their internetwork coupling for efficient mutual synchronization
based on the phase reduction theory developed in Ref. [H. Nakao, S. Yasui, M.
Ota, K. Arai, and Y. Kawamura, Chaos 28, 045103 (2018)]. The dynamical
equations describing a pair of weakly coupled networks are reduced to a pair of
coupled phase equations, and the linear stability of the synchronized state
between the networks is represented as a function of the internetwork coupling
matrix. We seek the optimal coupling by minimizing the Frobenius and L1 norms
of the internetwork coupling matrix for the prescribed linear stability of the
synchronized state. Depending on the norm, either a dense or sparse
internetwork coupling yielding efficient mutual synchronization of the networks
is obtained. In particular, a sparse yet resilient internetwork coupling is
obtained by L1-norm optimization with additional constraints on the individual
connection weights.
