Finite plateau in spectral gap of polychromatic constrained random networks

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.