Controllability of complex networks via pinning

We study the problem of controlling a general complex network toward an assigned synchronous evolution by means of a pinning control strategy. We define the pinning controllability of the network in terms of the spectral properties of an extended network topology. The roles of the control and coupling gains, as well as of the number of pinned nodes, are also discussed.

Figure 1

(Color online) Left pictures: Combinatorial Laplacian (symmetric topology). (a) Behavior of the eigenratio ${\mu }_{N+1}∕{\mu }_2$ of the Laplacian spectrum as function of the control gain $\kappa$ for a BA network, with average degree $⟨k⟩=4$, $p=0.1$, $\sigma =0.30$. (b) Control error at regime $E$ as function of the control gain $\kappa$ under the same conditions as in the upper plot. Right pictures: Normalized Laplacian (asymmetric topology). (c) $R^{N+1}$ vs $\kappa$ for a BA network, with average degree $⟨k⟩=4$, $p=0.1$, $\sigma =2.8$. (d) $E$ vs $\kappa$ under the same conditions as in the upper plot. In (a) and (c) the horizontal continuous lines represent the eigenratio $R^N$ of the corresponding uncontrolled networks.

Figure 2

(Color online) Curve level sets of the natural logarithm of $E$, asymptotic value of the control error, as a function of both $\kappa$ and $\sigma$, for a BA network of ${10}^3$ nodes, with average degree $⟨k⟩=4$, $p=0.1$. The left (right) panel shows the case of combinatorial (normalized) Laplacian.

Figure 3

(Color online) Left pictures: Combinatorial Laplacian (symmetric topology). (a) Behavior of the eigenratio ${\mu }_{N+1}∕{\mu }_2$ of the Laplacian spectrum as function of the control gain $\kappa$ for a BA network, with average degree $⟨k⟩=4$, $\sigma =0.30$, $\kappa =10$. (b) Control error at regime $E$ as function of the control gain $\kappa$ under the same conditions as in the upper plot. Right pictures: Normalized Laplacian (asymmetric topology). (c) $R^{N+1}$ vs $\kappa$ for a BA network, with average degree $⟨k⟩=4$, $\sigma =2.8$, $\kappa =1.5$. (d) $E$ vs $\kappa$ under the same conditions as in the upper plot. In all the plots the legend is as follows: squares (triangles) are used in the case of random (selective) pinning. In (a) and (c) the horizontal continuous lines represent the eigenratio $R^N$ of the corresponding uncontrolled networks.