Tweaking synchronization by connectivity modifications

Natural and man-made networks often possess locally treelike substructures. Taking such tree networks as our starting point, we show how the addition of links changes the synchronization properties of the network. We focus on two different methods of link addition. The first method adds single links that create cycles of a well-defined length. Following a topological approach, we introduce cycles of varying length and analyze how this feature, as well as the position in the network, alters the synchronous behavior. We show that in particular short cycles can lead to a maximum change of the Laplacian's eigenvalue spectrum, dictating the synchronization properties of such networks. The second method connects a certain proportion of the initially unconnected nodes. We simulate dynamical systems on these network topologies, with the nodes' local dynamics being either discrete or continuous. Here our main result is that a certain number of additional links, with the relative position in the network being crucial, can be beneficial to ensure stable synchronization.

Figure 1

Sketch of a balanced tree network $G(m,3)$. In the single link addition procedure, we distinguish two cases, namely connecting nodes in the same fundamental branch (“in,” $▄▄$) or in different ones(“out,” $▄▖▄$), i.e., every path between the nodes before the addition contains the root node.

Figure 2

(a) Minimal and (b) maximal increase $\mathrm{\Delta }{\lambda }_{\max }$ of the largest Laplacian eigenvalue ${\lambda }_{\max }$ for various wiring choices. For reference, the integers denominate the length of the (a) shortest and (b) longest cycle possible for each configuration. For the link classification “in” vs “out,” see Fig. 1.

Figure 3

Increase of the maximum Laplacian eigenvalue ${\lambda }_{\max }$ for connections (a) on the same level creating cycles of length 3 and (b) between different levels creating cycles of length 4.

Figure 4

The average change of the maximum Laplacian eigenvalue is given as bullets while the shaded area indicates the width of the distribution between the minimal and the maximal change within the possible cycles in the network.

Figure 5

Relative occurrence of cycle lengths when randomly connecting two nodes in a $G(5,3)$ balanced tree.

Figure 6

Coupled logistic map in (a) periodic and (b) chaotic regimes, and Rössler oscillators in (c) periodic and (d) chaotic regimes. The upper plots depict the nonzero eigenvalues of the Laplacian $\mathbf{L}$ for different probability $p$ of link addition (see the text for details). The shaded area denotes the existence of stable synchronous solutions. In all panels, the initial topology was $G(3,3)$.

Figure 7

Pictured are the average eigenvalues ${\lambda }_2$ and ${\lambda }_{max}$ for varying linkage probability $p$; the shading indicates one standard deviation. The inset magnifies the regime of $0<p<0.02$. Note that the two eigenvalues ${\lambda }_2$ and ${\lambda }_{max}$ bound the eigenvalue spectrum of the Laplacian.