Spectral properties of Supra-Laplacian for partially interdependent networks.

The spectrum of the Laplacian matrices of complex networks is a key factor in network functionality. In this paper, the spectral properties of Supra-Laplacian for partially interdependent networks are investigated. Based on Matrix Perturbation Theory, refined results of the eigenvalue properties of Laplacian matrices are provided, which shows that the size relationship of the first-order approximate solutions of the eigenvalues remains unchanged, even if there is perturbation. Using these results, the theoretical approximate formulae of the minimum non-zero eigenvalue and the maximum eigenvalue of Supra-Laplacian for partially interdependent networks are derived, respectively. The outcomes in this paper are more general and need fewer calculations than that in the literature [13], [14]. Finally, the simulations and applications of synchronizability and diffusion process verify the feasibility and effectiveness of the proposed solutions. The findings can be instructive for networks when linking to the spectral properties of the Supra-Laplacian.