Spectral algorithms have been shown to be powerful tools for a variety of tasks in network science. In this talk, we will review how the spectral properties of certain algebraic representations of a network, such as the adjacency matrix and the Laplacian, are related to structural features of a network. Specifically, we will focus on how (almost) invariant subspaces can be induced by the network structure, and how these insights can be employed for network analysis.

To illustrate our ideas we discuss a number of application scenarios such as the analysis of nonlinear oscillator networks, and the inference of hierarchical stochastic block-models from network data.