  Network topology inference is a cornerstone problem in statistical analyses
of complex systems. In this context, the fresh look advocated here permeates
benefits from convex optimization and graph signal processing, to identify the
so-termed graph shift operator (encoding the network topology) given only the
eigenvectors of the shift. These spectral templates can be obtained, for
example, from principal component analysis of a set of graph signals defined on
the particular network. The novel idea is to find a graph shift that while
being consistent with the provided spectral information, it endows the network
structure with certain desired properties such as sparsity. The focus is on
developing efficient recovery algorithms along with identifiability conditions
for two particular shifts, the adjacency matrix and the normalized graph
Laplacian. Application domains include network topology identification from
steady-state signals generated by a diffusion process, and design of a graph
filter that facilitates the distributed implementation of a prescribed linear
network operator. Numerical tests showcase the effectiveness of the proposed
algorithms in recovering synthetic and structural brain networks.
