Enhancing synchronization by directionality in complex networks

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.

💡 Research Summary

The paper addresses the problem of improving synchronizability in complex networks by assigning directions to links while keeping the underlying topology and link weights unchanged. Previous work introduced the Residual Degree Gradient (RDG) method, which selects nodes with the smallest residual degree and orients a prescribed number of incident edges toward them. Although RDG can increase synchronizability in many cases, the authors demonstrate that it may create incommunicable components: a cut‑vertex can end up with only incoming links, cutting off information flow between subgraphs. In such situations the second smallest real Laplacian eigenvalue λ₂ʳ becomes zero, yielding a synchronizability index R = λ₂ʳ/λ_Nʳ = 0, i.e., complete synchronization is impossible.

To overcome this limitation, the authors propose the Residual Edge‑Betweenness Gradient (REBG) algorithm. For each node i they define a score
s_i = Σ_j a_ij · (l_ij)^θ,
where a_ij indicates the presence of an undirected edge, l_ij is the edge betweenness of (i,j) computed on the original undirected graph, and θ∈