Eigenvalue spectra of modular networks

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.