  We study structure, eigenvalue spectra and diffusion dynamics in a wide class
of networks with subgraphs (modules) at mesoscopic scale. The networks are
grown within the model with three parameters controlling the number of modules,
their internal structure as scale-free and correlated subgraphs, and the
topology of connecting network. Within the exhaustive spectral analysis for
both the adjacency matrix and the normalized Laplacian matrix we identify the
spectral properties which characterize the mesoscopic structure of sparse
cyclic graphs and trees. The minimally connected nodes, clustering, and the
average connectivity affect the central part of the spectrum. The number of
distinct modules leads to an extra peak at the lower part of the Laplacian
spectrum in cyclic graphs. Such a peak does not occur in the case of
topologically distinct tree-subgraphs connected on a tree. Whereas the
associated eigenvectors remain localized on the subgraphs both in trees and
cyclic graphs. We also find a characteristic pattern of periodic localization
along the chains on the tree for the eigenvector components associated with the
largest eigenvalue equal 2 of the Laplacian. We corroborate the results with
simulations of the random walk on several types of networks. Our results for
the distribution of return-time of the walk to the origin (autocorrelator)
agree well with recent analytical solution for trees, and it appear to be
independent on their mesoscopic and global structure. For the cyclic graphs we
find new results with twice larger stretching exponent of the tail of the
distribution, which is virtually independent on the size of cycles. The
modularity and clustering contribute to a power-law decay at short return
times.
