Centrality analysis—that is, identifying the most important nodes in a network, or graph—has diverse applications ranging from web search (e.g., PageRank) to the control of dynamics on networks (e.g., oscillator synchronization and information dissemination). Networks are commonly encoded by matrices (e.g., adjacency or Laplacian), and their spectral decompositions are widely utilized. Matrix perturbation theory provides a formalism to analyze the impact of structural perturbations on spectral-based network analyses, including—but not limited to—eigenvector-based centrality measures, such as Bonacich’s eigenvector centrality, Kleinberg’s hub/authority scores, and Google’s PageRank algorithm. First, I will leverage classical results for nonsingular spectral perturbations to quantify the importances of nodes, edges, and subgraph motifs in network-coupled dynamical systems—a topic closely related to eigenvalue/eigenvector elasticity for perturbed dynamical systems. For heterogeneous dynamical systems, we find it crucial to take into account both the structural heterogeneity and dynamics’ heterogeneity. Turning to data-science applications, I will introduce an approach using perturbation theory for Von Neumann entropy to quantify the extent to which a given empirical network belongs to a random-graph (null-model) ensemble. I will conclude with ongoing work in which we develop singular perturbation theory for multimodal datasets represented by block-structured “supra” matrices (akin to unfolded tensors). I will introduce a generalization of eigenvector-based centralities for multiplex and temporal networks (wherein layers represent complementary data sources and/or different time instances) and analyze the limiting behaviors under very strong and weak interlayer coupling.