  We proposed a method called residual edge-betweenness gradient (REBG) to
enhance synchronizability of networks by assignment of link direction while
keeping network topology and link weight unchanged. Direction assignment has
been shown to improve the synchronizability of undirected networks in general,
but we find that in some cases incommunicable components emerge and networks
fail to synchronize. We show that the REBG method can effectively avoid the
synchronization failure ($R=\lambda_{2}^{r}/\lambda_{N}^{r}=0$) which occurs in
the residual degree gradient (RDG) method proposed in Phys. Rev. Lett. 103,
228702 (2009). Further experiments show that REBG method enhance
synchronizability in networks with community structure as compared with the RDG
method.
