The stability (or instability) of synchronization is important in a number of real world systems, including the power grid, the human brain and biological cells. For identical synchronization, the synchronizability of a network, which can be measured by the range of coupling strength that admits stable synchronization, can be optimized for a given number of nodes and links. Depending on the geometric degeneracy of the Laplacian eigenvectors, optimal networks can be classified into different sensitivity levels, which we define as a networks sensitivity index. We introduce an efficient and explicit way to construct optimal networks of arbitrary size over a wide range of sensitivity and link densities. Using coupled chaotic oscillators, we study synchronization dynamics on optimal networks, showing that cospectral optimal networks can have drastically different speed of synchronization. Such difference in dynamical stability is found to be closely related to the different structural sensitivity of these networks: generally, networks with high sensitivity index are slower to synchronize, and, surprisingly, may not synchronize at all, despite being theoretically stable under linear stability analysis.
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