Synchronizing Distant Nodes: A Universal Classification of Networks

Stability of synchronization in delay-coupled networks of identical units generally depends in a complicated way on the coupling topology. We show that for large coupling delays synchronizability relates in a simple way to the spectral properties of the network topology. The master stability function used to determine the stability of synchronous solutions has a universal structure in the limit of large delay: It is rotationally symmetric around the origin and increases monotonically with the radius in the complex plane. This allows a universal classification of networks with respect to their synchronization properties and solves the problem of complete synchronization in networks with strongly delayed coupling.

Figure 1

(a) Pseudocontinuous spectrum $\delta (\omega )$ (lines) and location of the exact roots (crosses) for a one-dimensional complex map for $r=3.3>r_0=3$ and $r=2.7<r_0=3$. Parameters: $A=0.4$, $B=0.2$, $\psi =0$, and $\tau =30$. (b) Contour line ${\lambda }_{\max }=0$ of the MSF for coupled semiconductor lasers according to Eq. (12) for delay times $\tau =1$ (solid), 8 (dashed), 20 (dash-dotted), and 1000 (dotted).