Perturbation analysis and comparison of network synchronization methods

In many networked systems, synchronization is important and useful, and how to enhance synchronizability is an interesting problem. Based on the matrix perturbation theory, we analyze five methods of network synchronization enhancement, including the link removal, node removal, dividing hub node, pull control, and pinning control methods, and obtain explicit expressions for eigenvalue changes. By these comparisons, we find that, among all these methods, the pull control method is remarkable, as it can extend the synchronization (coupling strength) region from both the left and right sides, for any controlled node. Extensive simulation results are given to support the accuracy of the perturbation-based analysis.

Figure 1

Schematic show of network synchronization control under different methods, including (a) removing a link between nodes $k$ and $m$ [15], (b) removing a node, $k$ [45], (c) dividing the node $k_0$ into three nodes $k_0$, $k_1$, and $k_2$ [17], (d) pull control method on the node $k$ [30], and (e) pinning control method on the node $k$ [25, 26]. See more details in the text.

Figure 2

Results for the link removal method on the unweighted small-world network. (a) The numerical results (black line) and estimates (pink line) of the eigenvalue change $\mathrm{\Delta }{\lambda }_2$ for one link removal, to show a perfect match. (c) The estimate of $\mathrm{\Delta }{\lambda }_2$ plotted against its numerical result, to show an approximately linear relation. (b), (d) Similar to (a) and (c) but for $\mathrm{\Delta }{\lambda }_N$ instead. Here the small-world network parameters are the network size $N=200$, the average degree $\langle k\rangle =12$, and the rewiring probability $p=0.3$.

Figure 3

Similar to Fig. 2 but for the unweighted Erdos-Renyi network [the left four panels: (a), (b), (e), and (f)] and the unweighted scale-free network [the right four panels: (c), (d), (g), and (h)]. The straight lines in (e)–(h) are linear fits of the data, to guide an eye. Clearly they show that the theoretic prediction matches with the numerical results well. Here the network parameters are set as $N=200$ and $\langle k\rangle =12$.

Figure 4

Results for link removal on weighted networks. (a), (b), (c) The estimates of the eigenvalue change $\mathrm{\Delta }{\lambda }_2$ plotted versus their numerical results for the SW, ER, and SF networks, respectively. (d), (e), (f) Similar to (a), (b), and (c) but for the eigenvalue change $\mathrm{\Delta }{\lambda }_N$ instead. Clearly now $\mathrm{\Delta }{\lambda }_2$ and $\mathrm{\Delta }{\lambda }_N$ can be either positive or negative, which make the signs of $\mathrm{\Delta }{\varepsilon }_1$ and $\mathrm{\Delta }{\varepsilon }_2$ for all links uncertain, and the results are similar for all different types of complex networks. Here the network parameters are $N=200$ and $\langle k\rangle =12$ for all networks and $p=0.3$ for the SW network.

Figure 5

Results for the node removal on unweighted networks. (a), (b), (c) The estimates of $\mathrm{\Delta }{\lambda }_2$ versus numerical results for the SW, ER, and SF networks, respectively. (d), (e), (f) Similar to (a)–(c), but for the results of $\mathrm{\Delta }{\lambda }_N$ instead. Similar to the link removal on weighted networks in Fig. 4; here both $\mathrm{\Delta }{\lambda }_2$ and $\mathrm{\Delta }{\lambda }_N$ can be either positive or negative. The straight lines in the figures are linear fits of the data, to guide an eye. Here the network parameters are $N=200$, $\langle k\rangle =12$, and $p=0.3$.

Figure 6

Similar to Fig. 5 but for the weighted SW, ER, and SF networks instead.

Figure 7

Similar to Fig. 5, but for the hub dividing method instead. Now the values of $\mathrm{\Delta }{\lambda }_2$ are all negative, and the values of most of $\mathrm{\Delta }{\lambda }_N$ are around zero (being either positive or negative) and some of them are extremely smaller. (g), (h), (i) Plots of the numerical result of $\mathrm{\Delta }{\lambda }_N$ as a function of the degree of removed node, indicating that the smallest value of $\mathrm{\Delta }{\lambda }_N$ corresponds to the hub node with the largest degree. Here we use the network parameters $N=200$, $p=0.3$, and $\langle k\rangle =4$, which are smaller to display the hub effect.

Figure 8

Similar to Fig. 5, but for the pull control method instead. Clearly now the values of $\mathrm{\Delta }{\lambda }_2$ are all positive and those of $\mathrm{\Delta }{\lambda }_N$ are all negative. Based on these observations, the network synchronization region can expand from both the left and right coupling strength sides, ${\varepsilon }_1$ and ${\varepsilon }_2$. Therefore, the pull synchronization enhancement method is remarkable. Here the network parameters are $N=200$, $\langle k\rangle =12$, and $p=0.3$.

Figure 9

Controlling results for the pinning control method. (a), (b), (c) Plots of the eigenvalue change $\mathrm{\Delta }{\lambda }_N$ for different controlled nodes on the SW, ER, and SF networks, respectively. The estimate (pink dotted line) coming from the analysis in Eq. (44) matches well with the numerical results (blue open circles). (d), (e), (f) The results for $\mathrm{\Delta }{\lambda }_1$ including the analytic results of Eq. (47) instead. Here we set $N=200$, $\langle k\rangle =12$, and $p=0.3$.