Eigenvector perturbations of complex networks

• We derive a formula to compute eigenvalue perturbation by eigenvector perturbation.
• Two eigenvector perturbation policies and their respective definitions are given.
• We derive the theoretical allowed ranges of two eigenvector perturbation factors.
• The theoretic allowed values of perturbation factors are validated by experiments.
• We seek out eigenvector perturbation forms equivalent to topological perturbations.

Recently spectral perturbations, involving eigenvalue and eigenvector perturbations, which have attracted more attentions than conventional topological perturbations, are used to analyze and promote the robustness of complex networks. However, to the best of our knowledge, the study of eigenvector perturbation and the equivalence between it and topological perturbation has not been found yet. In this paper, we first deduce the mathematical relationship between eigenvalue perturbation and its corresponding eigenvector perturbation for network reconstructions. Afterwards, two perturbed forms of eigenvector spectrum, global perturbation and local perturbation, are defined, such that we can examine the impacts of eigenvector perturbations on network robustness, and compare those to the impacts of topological perturbations on robustness. Meanwhile, the theoretical ranges of the allowed values of two eigenvector perturbation factors are derived in terms of the accurate reconstruction condition of networks, and validated by experimental simulations. By comparison our finding is that the eigenvector perturbations we define seem equivalent to topological perturbations.