Abstract
We present an adaptation of the Kato--Temple inequality for bounding perturbations of eigenvalues with applications to statistical inference for random graphs, specifically hypothesis testing and change-point detection. We obtain explicit high-probability bounds for the individual distances between certain signal eigenvalues of a graph's adjacency matrix and the corresponding eigenvalues of the model's edge probability matrix, even when the latter eigenvalues have multiplicity. Our results extend more broadly to the perturbation of singular values in the presence of quite general random matrix noise.
Highlights
Eigenvalues and eigenvectors are structurally fundamental quantities associated with matrices and are widely studied throughout mathematics, statistics, and engineering disciplines
In the context of certain random graph models, the eigenvalues and eigenvectors associated with the underlying matrix-valued model parameter, the edge probability matrix, exhibit similar information
We present our main results for the inhomogeneous Erdos–Renyi model (IERM) setting
Summary
Eigenvalues and eigenvectors are structurally fundamental quantities associated with matrices and are widely studied throughout mathematics, statistics, and engineering disciplines. Given an observed graph, the eigenvalues and eigenvectors of associated matrix representations (such as the adjacency matrix or Laplacian matrix) encode structural information about the graph (e.g. community structure, connectivity [8]). In the context of certain random graph models, the eigenvalues and eigenvectors associated with the underlying matrix-valued model parameter, the edge probability matrix, exhibit similar information. It is natural to study how “close” the eigenvalues and eigenvectors of a graph are to the underlying model quantities. J. Cape et al./Kato-Temple inequality, eigenvalue concentration, and graph inference 1
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
