Laplacian spectral properties of complex networks

We consider spectral analysis of complex networks. It is well known that the eigenvalue spectrum of complex networks provides information about their structural properties. Therefore, we present spectral properties of some different real-world networks such as regular networks, random networks, small-world networks, scale-free networks, and so on. We find that in random networks, the smallest nonzero eigenvalue grows approximately linearly with respect to the probability p. As a result of this, some estimates for the smallest nonzero eigenvalues of random networks can be obtained. More interestingly, it is shown a strong correlation between the eigenvalue spectrum and degree sequence in the networks, especially in scale-free networks. Making use of this correlation, we develop a local algorithm to determine the eigenvalue λ_i+1 from λ_i.