Dynamics and Directionality in Complex Networks

We investigate how we can improve the synchronizability of complex networks simply by changing the link direction while conserving the local link weights and topology. Performing the linear stability analysis of synchronization and numerical simulation of the Kuramoto model in the directed networks, we find that while a random assignment of link directions generally weakens the degree of synchronization, a properly organized directionality can systematically enhance the network synchronization. In this respect, we suggest a simple method of changing the link direction according to the larger residual degree starting from small residual degree nodes. This result provides plausible applications to control the synchronizability of systems in various fields.

Figure 1

Illustration of the construction of RDG network. (1) Regarding a given network as undirected, each node $i$ is given degree $k_i$ and the mean degree is defined as $\overline{k}=\sum _{}^{}{k}_i/N$. We then introduce a variable, residual degree, ${\tilde{k}}_i$ for each node $i$, which represents the number of remaining links as yet without a direction, (a) initially assigned as ${\tilde{k}}_i=k_i$. (2) The smallest $\tilde{k}$ node is selected among the remaining nodes, and the outgoing directed links are assigned [15] from the selected node to its neighbors through the residual links until the number of outgoing links reaches half of the mean degree, i.e., $\lceil \overline{k}/2\rceil$, or the selected node exhausts all links to its neighbors. If there are multiples of the same rank $\tilde{k}$, we choose the smaller $k$ node first. (3) The residual degree of the selected node is reduced by the number of drawn outgoing links. Furthermore, each of its connected nodes $l$ reduces its residual degree ${\tilde{k}}_l$ by one since it has an incoming link. (4) Steps (2) and (3) are repeated until all nodes and links are exhausted (b)–(e) [17].

Figure 2

The tendency of the smallest nonzero and the largest real part of eigenvalues, and their ratio. As $\alpha$ increases from zero to one, the smallest real part of the eigenvalue of the RDG network (square) increases (a) and the largest real part decreases (b). Therefore, the real part eigenvalue ratio becomes close to 1 compared to that of RAD, which means the synchronizability is enhanced (c). The RAD network (circle) shows opposite behaviors. The error bars represent the standard deviation from network ensembles.

Figure 3

Dynamic behavior of Kuramoto synchrony for RAD and RDG networks. (a) The RAD network shows that as $\alpha$ increases, ${\Delta }_{\mathrm{st}}$ decreases and synchronization becomes slower. On the other hand, (b) the RDG network shows that as the network becomes more directed, ${\Delta }_{\mathrm{st}}$ increases and the system is synchronized more quickly.

Figure 4

The outgoing degree distribution of RAD and RDG networks. (a) The range of outgoing degree distribution becomes broad as $\alpha$ increases. (b) The outgoing degree distribution of RDG broadens in the early stages, but narrows immediately after $\alpha$ passes 0.4.

Figure 5

The tendency of the smallest nonzero and the largest real eigenvalues, and the real eigenvalue ratio for the Barabási-Albert network. We used the parameter of three outgoing links for each new node starting from the initial condition of connected three nodes like a triangle.