Abstract
Complex real-world phenomena are often modeled as dynamical systems on networks. In many cases of interest, the spectrum of the underlying graph Laplacian sets the system stability and ultimately shapes the matter or information flow. This motivates devising suitable strategies, with rigorous mathematical foundation, to generate Laplacians that possess prescribed spectra. In this paper, we show that a weighted Laplacian can be constructed so as to exactly realize a desired complex spectrum. The method configures as a non trivial generalization of existing recipes which assume the spectra to be real. Applications of the proposed technique to (i) a network of Stuart–Landau oscillators and (ii) to the Kuramoto model are discussed. Synchronization can be enforced by assuming a properly engineered, signed and weighted, adjacency matrix to rule the pattern of pairing interactions.
Highlights
Complex networks play a role of paramount importance for a wide range of problems, of cross-disciplinary breadth
For reaction-diffusion systems defined on networks, the stability of the inspected equilibrium is dictated by the spectrum of the discrete Laplacian matrix [1,2,3]
Stability is an attribute of paramount importance as it relates to resilience, the ability of a given system to oppose to external perturbations that would take it away from the existing equilibrium
Summary
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Keywords: directed network, diffusion, eigenvalues and eigenvectors, Laplacian operator, linear stability analysis
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