We study the structure characteristics of complex networks from the representative eigenvectors of the adjacent matrix. The probability distribution function of the components of the representative eigenvectors are proposed to describe the localizations on networks where the Euclidean distance is invalid. Several quantities are used to describe the localization properties of the representative states, as the participation ratio, the structural entropy, the probability distribution function of the nearest neighbor level spacings for spectra of complex networks. The whole cellular networks in real world, the Watts-Strogatz small world and Barabasi-Albert scale-free networks are considered. The networks have nontrivial localization properties due to the nontrivial topological structures. It is found that the ascend-ranked series of the occurring probabilities at the nodes behave generally multi-fractal. This characteristic can be used as a structure measure of complex networks.

In a complex network, different groups of nodes may have existed for different amounts of time. To detect the evolutionary history of a network is of great importance. We present a spectral-analysis based method to address this fundamental question in network science. In particular, we find that there are complex networks in the real-world for which there is a positive correlation between the eigenvalue magnitude and node age. In situations where the network topology is unknown but short time series measured from nodes are available, we suggest to uncover the network topology at the present (or any given time of interest) by using compressive sensing and then perform the spectral analysis. Knowledge of ages of various groups of nodes can provide significant insights into the evolutionary process underpinning the network. It should be noted, however, that at the present the applicability of our method is limited to the networks for which information about the node age has been encoded gradually in the eigen-properties through evolution.

We study spectra of directed networks with inhibitory and excitatory couplings. We investigate in particular eigenvector localization properties of various model networks for different values of correlation among their entries. Spectra of random networks with completely uncorrelated entries show a circular distribution with delocalized eigenvectors, whereas networks with correlated entries have localized eigenvectors. In order to understand the origin of localization we track the spectra as a function of connection probability and directionality. As connections are made directed, eigenstates start occurring in complex-conjugate pairs and the eigenvalue distribution combined with the localization measure shows a rich pattern. Moreover, for a very well distinguished community structure, the whole spectrum is localized except few eigenstates at the boundary of the circular distribution. As the network deviates from the community structure there is a sudden change in the localization property for a very small value of deformation from the perfect community structure. We search for this effect for the whole range of correlation strengths and for different community configurations. Furthermore, we investigate spectral properties of a metabolic network of zebrafish and compare them with those of the model networks.