Network synchronization: optimal and pessimal scale-free topologies

By employing a recently introduced optimization algorithm we construct optimally synchronizable (unweighted) networks for any given scale-free degree distribution. We explore how the optimization process affects degree?degree correlations and observe a generic tendency toward disassortativity. Still, we show that there is not a one-to-one correspondence between synchronizability and disassortativity. On the other hand, we study the nature of optimally un-synchronizable networks, that is, networks whose topology minimizes the range of stability of the synchronous state. The resulting 'pessimal networks' turn out to have a highly assortative string-like structure. We also derive a rigorous lower bound for the Laplacian eigenvalue ratio controlling synchronizability, which helps understanding the impact of degree correlations on network synchronizability.

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