  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 network's 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.
