  We consider the canonical ensemble of multilayered constrained Erdos-Renyi
networks (CERN) and regular random graphs (RRG), where each layer represents
graph vertices painted in a specific color. We study the critical behavior in
such networks under changing the fugacity, $\mu$, which controls the number of
monochromatic triads of nodes. The behavior of considered systems is
investigated via the spectral properties of the adjacency and Laplacian
matrices of corresponding networks. For some wide region of $\mu$ we find the
formation of a finite plateau in the number of the intercolor links, which
exactly matches the finite plateau for the algebraic connectivity of the
network (the value of the first non-vanishing eigenvalue of the Laplacian
matrix, $\lambda_2$). We claim that at the plateau the restoring of the
spontaneously broken $Z_2$ symmetry by the mechanism of modes collectivization
in clusters of different colors occurs. The phenomena of a finite plateau
formation holds for the polychromatic (multilayer) networks with $M>2$ colors.
