Scaling and distribution of the width in regular and small-world synchronization networks in 2D*

We study the evolution, the scaling and the steady-state distribution of the width in two-dimensional regular and small-world (SW) networks motivated by a synchronization problem in distributed computing \footnote[2]{G. Korniss et al.,\textit{Science} \textbf{299}, 677 (2003).} \footnote[3]{H. Guclu and G. Korniss, \textit{Phys. Rev. E} \textbf{69} 065104(R) (2004).}. We find that in the regular network the system exhibits Kardar-Parisi-Zhang (KPZ) type roughening (de-synchronized state) with a very slow convergence to the KPZ width distribution. When SW links are added to the regular network one obtains a finite width in the thermodynamic limit (synchronized state). The distribution of the width in the SW network, however, is of non-Gaussian type with an exponential tale.