Adaptive Synchronization of Dynamics on Evolving Complex Networks

We study the problem of synchronizing a general complex network by means of an adaptive strategy in the case where the network topology is slowly time varying and every node receives at each time only one aggregate signal from the set of its neighbors. We introduce an appropriately defined potential that each node seeks to minimize in order to reach or maintain synchronization. We show that our strategy is effective in tracking synchronization as well as in achieving synchronization when appropriate conditions are met.

Figure 1

The plots show the time evolutions from $t=0$ to $t=100$ of $x_{i1}(t)$, $i=1,\dots ,N$, while the network topology evolves according to (17) in the case where the adaptation (16, 18) was implemented (a) and in the case where it was not implemented (b). Here the oscillators start synchronized, $\nu =1/(2{\tau }_s)$, $\gamma =2$. The network parameters are as follows: $N=50$, $⟨k⟩=10$.

Figure 2

${\overline{E}}_{x_1}$ calculated between $t_1=500$ and $t_2=1000$ vs $c$; $N=100$, $⟨k⟩=20$, $\nu =1/(2{\tau }_s)$, $\gamma =2$, $B_i(0)=1$. The dashed line represents ${\overline{E}}_{x_1}$ vs $c$ for the case $B_i(0)=3\times {10}^4$.