Repeated-drive adaptive feedback identification of network topologies

The identification of the topological structures of complex networks from dynamical information is a significant inverse problem. How to infer the information of network topology from short-time dynamical data is a challenging topic. The presence of synchronization among nodes makes the identification of network topology difficult. In this paper we present an efficient method called the repeated-drive adaptive feedback scheme to reveal the network connectivity from short-time dynamics. By applying the short asynchronous transient data as a repeated drive, the adjacency matrix can be successfully determined in terms of the modified adaptive feedback scheme. This improved scheme is valid for both synchronous and asynchronous cases of the network and is especially efficient in the presence of global or local synchronization, where the transient drive can be obtained by perturbing the system to get a very short asynchronous transient. The detection speed of our scheme exhibits the optimized effect by adjusting the time-series segment length and the coupling strength among nodes in the network.

Figure 1

(a) The topology of a network with four nodes. (b) The normalized global coupling strength $s\lambda$ against the master-stability function $\Psi \left(s\lambda \right)$ for a network of Lorenz oscillators. The curve of $\Psi \left(s\lambda \right)$ is below the dotted line $\Psi =0$ in the interval $s\lambda >0.65$.

Figure 2

(a) Top: The evolution of the difference in the first component of nodes 1 and 4, ${\Delta }_{1,4}\left(t\right)$, for the global coupling strength $s=1$. The network evolves from the asynchronous state to the synchronous state. Bottom: The evolution of one element in the estimator $d_{14}\left(t\right)$ driven by this transient time series, and the true value $c_{14}=1$ is labeled by dotted line. (b) The probability of successful identification, $P_s$, varies with the global coupling strength $s$ by using the traditional AFS.

Figure 3

The evolutions of different terms in the Lyapunov function [Eq. (3)] and the Lyapunov function $V\left(t\right)$ under the repeated asynchronous drives ($\Delta t=2.0$), where the global coupling strength $s=1$. At the end of every drive, we reset $t=0$. (a) $V_1\left(t\right)$, (b) $V_2\left(t\right)$, (c) $V_3\left(t\right)$, and (d) the Lyapunov function $V\left(t\right)$.

Figure 4

The numerical simulation results for the RDAFS, where the global coupling strength $s=1$. (a) Top: The repeated transient time series $\{{\mathbf{x}}_i\left(t\right),t\in (0,2)\}$ with each segment the same as in Fig. 2. Bottom: The evolution of $d_{14}\left(t\right)$, and $c_{14}=1$ is labeled by the dotted line. Coincidence can be clearly found. (b) The rate of successful identification, $P_s$, versus the global coupling strength $s$ for 2000 drives.

Figure 5

(a) The evolution of $d_{\text{err}}\left(t\right)$ for $s=0.01$ (asynchronous regime) and different segment lengths $\Delta t=0.2,2,20$. (b) The evolution of $d_{\text{err}}\left(t\right)$ for $s=4.0$ (global synchronization) and different segment lengths $\Delta t=0.2,2,10$. (c) The identification speed $v$ varying against the segment length $\Delta t$ for the global coupling $s=4.0$ (global synchronization). (d) The identification speed $v$ varying against the global coupling $s$ for different segment lengths $\Delta t=0.2,2,20$.