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Spectra of Laplacian Matrices of Weighted Graphs: Structural Genericity Propoerties

Please always quote using this URN:urn:nbn:de:0296-matheon-14045
  • This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity) and its associated eigenvector, the so-called Fiedler vector. Here we prove that, given a Laplacian matrix, it is possible to perturb the weights of the existing edges in the underlying graph in order to obtain simple eigenvalues and a Fiedler vector composed of only non-zero entries. These structural genericity properties with the constraint of not adding edges in the underlying graph are stronger than the classical ones, for which arbitrary structural perturbations are allowed. These results open the opportunity to understand the impact of structural changes on the dynamics of complex systems.
Metadaten
Author:Jan Philipp Pade, Camille Poignard, Tiago Pereira
URN:urn:nbn:de:0296-matheon-14045
Referee:Caren Tischendorf
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2017/05/11
Release Date:2017/05/11
Tag:Spectra of Graphs, Laplacians, Perturbations of Eigenvalues
Institute:Research Center Matheon
Humboldt-Universität zu Berlin
Project:C Energy and Materials (Production) / C-SE3 Stability analysis of power networks and power network models
MSC-Classification:05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Cxx Graph theory (For applications of graphs, see 68R10, 81Q30, 81T15, 82B20, 82C20, 90C35, 92E10, 94C15) / 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
15-XX LINEAR AND MULTILINEAR ALGEBRA; MATRIX THEORY / 15Bxx Special matrices / 15B57 Hermitian, skew-Hermitian, and related matrices
47-XX OPERATOR THEORY / 47Axx General theory of linear operators / 47A55 Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
Preprint Number:1124
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