  The importance of the balance in inhibitory and excitatory couplings in the
brain has increasingly been realized. Despite the key role played by
inhibitory-excitatory couplings in the functioning of brain networks, the
impact of a balanced condition on the stability properties of underlying
networks remains largely unknown. We investigate properties of the largest
eigenvalues of networks having such couplings, and find that they follow
completely different statistics when in the balanced situation. Based on
numerical simulations, we demonstrate that the transition from Weibull to
Fr\'echet via the Gumbel distribution can be controlled by the variance of the
column sum of the adjacency matrix, which depends monotonically on the
denseness of the underlying network. As a balanced condition is imposed, the
largest real part of the eigenvalue emulates a transition to the generalized
extreme value statistics, independent of the inhibitory connection probability.
Furthermore, the transition to the Weibull statistics and the small-world
transition occur at the same rewiring probability, reflecting a more stable
system.
