  The study of complex networks has been one of the most active fields in
science in recent decades. Spectral properties of networks (or graphs that
represent them) are of fundamental importance. Researchers have been
investigating these properties for many years, and, based on numerical data,
have raised a number of questions about the distribution of the eigenvalues and
eigenvectors. In this paper, we give the solution to some of these questions.
In particular, we determine the limiting distribution of (the bulk of) the
spectrum as the size of the network grows to infinity and show that the leading
eigenvectors are strongly localized. We focus on the preferential attachment
graph, which is the most popular mathematical model for growing complex
networks. Our analysis is, on the other hand, general and can be applied to
other models.
