  The eigenvalues of matrices representing the structure of large-scale complex
networks present a wide range of applications, from the analysis of dynamical
processes taking place in the network to spectral techniques aiming to rank the
importance of nodes in the network. A common approach to study the relationship
between the structure of a network and its eigenvalues is to use synthetic
random networks in which structural properties of interest, such as degree
distributions, are prescribed. Although very common, synthetic models present
two major flaws: (\emph{i}) These models are only suitable to study a very
limited range of structural properties, and (\emph{ii}) they implicitly induce
structural properties that are not directly controlled and can deceivingly
influence the network eigenvalue spectrum. In this paper, we propose an
alternative approach to overcome these limitations. Our approach is not based
on synthetic models, instead, we use algebraic graph theory and convex
optimization to study how structural properties influence the spectrum of
eigenvalues of the network. Using our approach, we can compute with low
computational overhead global spectral properties of a network from its local
structural properties. We illustrate our approach by studying how structural
properties of online social networks influence their eigenvalue spectra.

Social and Information Networks;Systems and Control;Dynamical Systems;Optimization and Control;Data Analysis, Statistics and Probability;Physics and Society