Network synchronization of groups

In this paper we study synchronized motions in complex networks in which there are distinct groups of nodes where the dynamical systems on each node within a group are the same but are different for nodes in different groups. Both continuous time and discrete time systems are considered. We initially focus on the case where two groups are present and the network has bipartite topology (i.e., links exist between nodes in different groups but not between nodes in the same group). We also show that group synchronous motions are compatible with more general network topologies, where there are also connections within the groups.

Figure 1

A simple five-nodes network.

Figure 2

(Color online) Random networks with $N_x=N_y=5\times {10}^2$ nodes. The spectrum of $Q$ is shown for three sets of values of $p_{xy}$ and $p_{yx}$. Yellow (light gray) is used for $p_{xy}=p_{yx}=0.05$; red (dark gray) for $p_{xy}=0.5$ and $p_{yx}=0.05$; black for $p_{xy}=p_{yx}=0.5$. The continuous line is used to represent the circle of unit radius centered at $(0,0)$. The eigenvalues $\lambda =\pm 1$ associated with perturbations in the synchronization manifold are denoted by solid black dots.

Figure 3

(Color online) Log-log plot of ${\lambda }_{max}∕C$ versus $N$, for different values of $p_{xy}$ ranging between $0.1$ and $0.9$ in steps of 0.1. The straight line has slope $-1∕2$. The inset shows $C$ versus $p$.

Figure 4

(Color online) The behavior of ${\lambda }_{max}$ as a function of $p$ and of $N$ for large random networks with real spectra ($Q=K^{-1}{Q}^{\prime }$ and $Q^{\prime }$ symmetric). (a) ${\lambda }_{max}$ versus $p$, for $N_x=N_y=5\times {10}^2$ and $N_y=5\times {10}^2$ nodes. (b) Log-log plot of ${\lambda }_{max}∕C$ versus $N$ for $p=0.1$ and $p=0.5$ showing that [as for case (i)] ${\lambda }_{max}$ scales as $N^{-1∕2}$ for large $N$ (the solid line has slope $-1∕2$).

Figure 5

The upper panel shows $E_x+E_y$ versus ${\sigma }_x$ for ${\sigma }_y=0.65$. The continuous thin line represents $E_x+E_y$ at $t=100$; the dashed line, $E_x+E_y$ at $t=200$; the thick continuous line, $E_x+E_y$ at $t=300$. The network parameters are as follows: $N_x=200$, $N_y=300$, and $p=0.5$. The lower panel shows the master stability function evaluated at $\lambda =0$ (continuous line), and at $\lambda ={\lambda }_{max}=0.13$ (dashed line) versus ${\sigma }_x$.

Figure 6

The master stability function associated with the system (27) as a function of the real parameter $\lambda$, varying between 0 and 1.

Figure 7

The network shown in Fig. 1 is modified as a function of $w$. The plot shows the sum of the values of the errors $E_x+E_y$ as a function of the corresponding ${\lambda }_{max}$.

Figure 8

The plot shows the transient evolutions $x_i\left(t\right)$, $i=1,\dots ,N_x$ and $y_{j(1,2)}\left(t\right)$, $j=1,\dots ,N_y$. The network parameters are as follows: $N_x=200$, $N_y=300$, and $p=0.05$.

Figure 9

The synchronization errors $E_x$ and $E_y$ versus ${\sigma }_{yy}$ for Eqs. (36, 37) for a random network with $N_x=200$, $N_y=300$, $p_{xy}=p_{yx}=0.1$, and $p_{yy}=0.15$.

Figure 10

The top three plots show the time evolutions $x_i\left(t\right)$, $i=1,\dots ,2$, and $y_{j(1,2)}\left(t\right)$, $j=1,\dots ,3$, for the bipartite network represented in Fig. 1. The equations are those in Eqs. (23, 24): The state of the systems at the final time $t^{*}=100$ is shown by asterisks. The bottom three plots show the state evolution of the network in the case when a bidirectional diffusive link with an associated coupling constant equal to $2$ is added between the two $x$ nodes. It is seen that the presence of the added link causes the network to synchronize.