Regulating synchronous states of complex networks by pinning interaction with an external node

To shed light on how biological and technological systems can establish or maintain a synchronous functioning, we address the problem of how to engineer an external pinning action on a network of dynamical units. In particular, we study the regulation of a network toward a synchronized behavior by means of a bidirectional interaction with an external node that leaves unchanged its inner parameters and architecture. We demonstrate that there are two classes of networks susceptible of being regulated into a synchronous motion and provide a simple method, for each one of them, to properly design a pinning sequence to achieve regulation. We also discuss how the obtained sequence can be compared with a topological ranking of the network nodes.

Figure 1

(Color online) Log-linear graph showing the behavior of ${\lambda }_2\left(n\right)$ during the regulation of a class II system for (a) SW and (b) SF networks (see text for details). As reference, the values of ${\lambda }_2^{ini}$ (full circles) and ratios ${\mu }_1/\sigma$ (dashed lines) with (a) $\sigma =0.07$ and (b) $\sigma =0.045$ are given.

Figure 2

(Color online) (a) and (b) Log-log behavior of ${\lambda }_2\left(n\right)$ (blue open circles) and ${\lambda }_{N+1}\left(n\right)$ (red open squares) during regulation of a (a) SW network [as in Fig. 1a] and (b) SF network [as in Fig. 1b] of class III systems. The values of ${\lambda }_2^{ini}$ (blue full circle) and ${\lambda }_N^{ini}$ (red full square) refer to those of ${\mathcal{G}}_0$, and the horizontal dashed lines are in correspondence with the limiting thresholds ${\mu }_1/\sigma$ and ${\mu }_2/\sigma$ with (a) $\sigma =0.07$ and (b) $\sigma =0.045$. Panels (c) and (d) are zooms of the ${\lambda }_2\left(n\right)$ behavior shown in (a) and (b), respectively, with vertical (c) dotted and (d) dash-dotted lines marking the initial and final values of $n$ for which the (c) SW and (d) SF networks are regulated. For comparison, ${\lambda }_2^c\left(n\right)$ is also reported in blue solid line as obtained from a ranking centrality-based pinning regulation.

Figure 3

(Color online) Log-linear plots of the averaged synchronization error $⟨e⟩$ (see text for definition) vs. $n$ during the regulation of the same (a) SW and (b) SF networks used in Fig. 2 following the pinning sequence optimizing $R\left(n\right)$ given by Eq. (2). Vertical (a) dotted and (b) dash-dotted lines are added to show the agreement with the predicted range of regulability depicted in Figs. 2c, 2d respectively.

Figure 4

(Color online) (a) Log-linear behavior of $R\left(n\right)$, given by Eq. (2), for the SW (diamonds) and SF (crosses) networks used in Figs. 2a, 2b. The vertical dotted (dash-dotted) lines denote the range of $n$ in which the SW (SF) network is synchronized due to regulation. (b) Log-linear behavior of the ratio $\frac{{\lambda }_{N+1}\left(n\right)}{{\lambda }_2\left(n\right)}$, where ${\lambda }_2\left(n\right)$ and ${\lambda }_{N+1}\left(n\right)$ are those obtained from the regulation process of the SW network of class III systems reported in Fig. 2a. The shaded region corresponds to regulating sequences whose eigenratio values are larger than ${\lambda }_N^{ini}/{\lambda }_2^{ini}$ (horizontal dashed line) at which ${\mathcal{G}}_0$ displays an asynchronous motion.