  By numerical simulations, we investigate the onset of synchronization of
networked phase oscillators under two different weighting schemes. In scheme-I,
the link weights are correlated to the product of the degrees of the connected
nodes, so this kind of networks is named as the weight-degree correlated (WDC)
network. In scheme-II, the link weights are randomly assigned to each link
regardless of the node degrees, so this kind of networks is named as the
weight-degree uncorrelated (WDU) network. Interestingly, it is found that by
increasing a parameter that governs the weight distribution, the onset of
synchronization in WDC network is monotonically enhanced, while in WDU network
there is a reverse in the synchronization performance. We investigate this
phenomenon from the viewpoint of gradient network, and explain the contrary
roles of coupling gradient on network synchronization: gradient promotes
synchronization in WDC network, while deteriorates synchronization in WDU
network. The findings highlight the fact that, besides the link weight, the
correlation between the weight and node degree is also important to the network
dynamics.
