Generating directed networks with prescribed Laplacian spectra

Sara Nicoletti, Timoteo Carletti, Duccio Fanelli, Giorgio Battistelli, Luigi Chisci

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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.

Original languageEnglish
Article number015004
Pages (from-to)015004
Number of pages18
JournalJournal of Physics: Complexity
Issue number1
Early online date30 Sep 2020
Publication statusPublished - 1 Oct 2020


  • networks reconstruction
  • Laplace matrix
  • spectrum
  • directed network
  • inverse problem
  • Directed network
  • Eigenvalues and eigenvectors
  • Linear stability analysis
  • Diffusion
  • Laplacian operator


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