  Recent studies have shown that synchronizability of complex networks can be
significantly improved by asymmetric couplings, and increase of coupling
gradient is always in favor of network synchronization. Here we argue and
demonstrate that, for typical complex networks, there usually exists an optimal
coupling gradient under which the maximum network synchronizability is
achieved. After this optimal value, increase of coupling gradient could
deteriorate synchronization. We attribute the suppression of network
synchronization at large gradient to the phenomenon of network breaking, and
find that, in comparing with sparsely connected homogeneous networks, densely
connected heterogeneous networks have the superiority of adopting large
gradient. The findings are supported by indirect simulations of eigenvalue
analysis and direct simulations of coupled nonidentical oscillator networks.
