We explore the synchronization behavior in the
interdependent system, where the 1D network is
ferromagnetically intercoupled to the Watts-Strogatz (WS)
small-world network. In the absence of the internetwork
coupling, the former network is well known not to exhibit
the synchronized phase at any finite coupling strength,
whereas the latter displays the mean-field transition.
Through an analytic approach based on the mean-field
approximation, it is found that for the weakly coupled and
thus nonsynchronized 1D network becomes a heavier burden for
the synchronization process of the WS network. As the
intracoupling in the 1D network becomes stronger, the more
enhanced partial synchronization in the 1D network makes the
burden lighter. Extensive numerical simulations confirm
these expected behaviors, while exhibiting a reentrant
behavior in the intermediate range of coupling strength. The
nonmonotonic change of the critical value of JII is also
compared with the result from the numerical renormalization
group calculation. |