  In a recent paper, wavelet analysis was used to perturb the coupling matrix
in an array of identical chaotic systems in order to improve its
synchronization. As the synchronization criterion is determined by the second
smallest eigenvalue $\lambda_2$ of the coupling matrix, the problem is
equivalent to studying how $\lambda_2$ of the coupling matrix changes with
perturbation. In the aforementioned paper, a small percentage of the wavelet
coefficients are modified. However, this result in a perturbed matrix where
every element is modified and nonzero. The purpose of this paper is to present
some results on the change of $\lambda_2$ due to perturbation. In particular,
we show that as the number of systems $n \to \infty$, perturbations which only
add local coupling will not change $\lambda_2$. On the other hand, we show that
there exists perturbations which affect an arbitrarily small percentage of
matrix elements, each of which is changed by an arbitrarily small amount and
yet can make $\lambda_2$ arbitrarily large. These results give conditions on
what the perturbation should be in order to improve the synchronizability in an
array of coupled chaotic systems. This analysis allows us to prove and explain
some of the synchronization phenomena observed in a recently studied network
where random coupling are added to a locally connected array. Finally we
classify various classes of coupling matrices such as small world networks and
scale free networks according to their synchronizability in the limit.
