  A large variety of dynamical processes that take place on networks can be
expressed in terms of the spectral properties of some linear operator which
reflects how the dynamical rules depend on the network topology. Often such
spectral features are theoretically obtained by considering only local node
properties, such as degree distributions. Many networks, however, possess
large-scale modular structures that can drastically influence their spectral
characteristics, and which are neglected in such simplified descriptions. Here
we obtain in a unified fashion the spectrum of a large family of operators,
including the adjacency, Laplacian and normalized Laplacian matrices, for
networks with generic modular structure, in the limit of large degrees. We
focus on the conditions necessary for the merging of the isolated eigenvalues
with the continuous band of the spectrum, after which the planted modular
structure can no longer be easily detected by spectral methods. This is a
crucial transition point which determines when a modular structure is strong
enough to affect a given dynamical process. We show that this transition
happens in general at different points for the different matrices, and hence
the detectability threshold can vary significantly depending on the operator
chosen. Equivalently, the sensitivity to the modular structure of the different
dynamical processes associated with each matrix will be different, given the
same large-scale structure present in the network. Furthermore, we show that,
with the exception of the Laplacian matrix, the different transitions coalesce
into the same point for the special case where the modules are homogeneous, but
separate otherwise.
