Robustness of Optimal Synchronization in Real Networks

Experimental studies can provide powerful insights into the physics of complex networks. Here, we report experimental results on the influence of connection topology on synchronization in fiber-optic networks of chaotic optoelectronic oscillators. We find that the recently predicted nonmonotonic, cusplike synchronization landscape manifests itself in the rate of convergence to the synchronous state. We also observe that networks with the same number of nodes, same number of links, and identical eigenvalues of the coupling matrix can exhibit fundamentally different approaches to synchronization. This previously unnoticed difference is determined by the degeneracy of associated eigenvectors in the presence of noise and mismatches encountered in real-world conditions.

Figure 1

Schematic of a single optoelectronic node. Each node is coupled to the rest of the network (not shown) through fiber-optic links.

Figure 2

Nonmonotonic behavior of synchronization properties. (a) A path from a fully connected network ($m=12$) to an optimal tree network ($m=3$). At each step, the link removed is indicated by a dashed line. (b) The eigenvalue spread $\sigma$ for the networks in (a). (c) Experimentally measured mean convergence rate to synchronization $\overline{\mu }$ and associated standard deviation (bars) for the same networks.

Figure 3

Differentiating behavior between sensitive and nonsensitive networks. (a) All optimal binary networks with 4 nodes and 3 links. Each network is labeled according to its geometric degeneracy $g_d$. (b) Experimentally measured $⟨\theta (t)⟩$ for sensitive (green, $g_d=3$) and nonsensitive (blue, $g_d=1$) configurations, where the coupling is enabled at $t=0$. (c) Numerical simulation of the same networks and conditions considered in (b). Inset: The difference $\Delta \mu$ between the decay exponents of the networks considered in (b) when simulated in the absence of mismatches, noises, and time delays, as a function of $\theta (t)$, regarded as a tunable initial synchronization error.

Figure 4

Dependence of synchronization properties on experimental parameters. Setting observational noise $\eta =0.06$, coupling mismatch $\zeta =0.01$, and time delay $\tau =1.5\mathrm{ms}$ (dashed lines), which approximate the values estimated from the experiment, we simulated the effect of varying these parameters one at a time. (a) The effect of $\eta$ on the ensemble mean synchronization floor ${\overline{\theta }}_0$; (b) the effect of $\eta$ on the standard deviation of the floor $\delta$; (c) the effect of $\zeta$ on $\delta$; and (d) the effect of $\tau$ on ${\overline{\theta }}_0$. The superscript $n$ ($s$) denotes the nonsensitive (sensitive) network.