  We study the level spacing distribution p(s) in the spectrum of random
networks. According to our numerical results, the shape of p(s) in the
Erdos-Renyi (E-R) random graph is determined by the average degree <k>, and
p(s) undergoes a dramatic change when <k> is varied around the critical point
of the percolation transition, <k>=1. When <k> > 1, the p(s) is described by
the statistics of the Gaussian Orthogonal Ensemble (GOE), one of the major
statistical ensembles in Random Matrix Theory, whereas at <k>=1 it follows the
Poisson level spacing distribution. Closely above the critical point, p(s) can
be described in terms of an intermediate distribution between Poisson and the
GOE, the Brody-distribution. Furthermore, below the critical point p(s) can be
given with the help of the regularised Gamma-function. Motivated by these
results, we analyse the behaviour of p(s) in real networks such as the
Internet, a word association network and a protein protein interaction network
as well. When the giant component of these networks is destroyed in a node
deletion process simulating the networks subjected to intentional attack, their
level spacing distribution undergoes a similar transition to that of the E-R
graph.
