  Inspired by the importance of inhibitory and excitatory couplings in the
brain, we analyze the largest eigenvalue statistics of random networks
incorporating such features. We find that the largest real part of eigenvalues
of a network, which accounts for the stability of an underlying system,
decreases linearly as a function of inhibitory connection probability up to a
particular threshold value, after which it exhibits rich behaviors with the
distribution manifesting generalized extreme value statistics. Fluctuations in
the largest eigenvalue remain somewhat robust against an increase in system
size but reflect a strong dependence on the number of connections, indicating
that systems having more interactions among its constituents are likely to be
more unstable.

Disordered Systems and Neural Networks