Maximum performance at minimum cost in network synchronization

Takashi Nishikawa*, Adilson E. Motter

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

123 Scopus citations


We consider two optimization problems on synchronization of oscillator networks: maximization of synchronizability and minimization of synchronization cost. We first develop an extension of the well-known master stability framework to the case of non-diagonalizable Laplacian matrices. We then show that the solution sets of the two optimization problems coincide and are simultaneously characterized by a simple condition on the Laplacian eigenvalues. Among the optimal networks, we identify a subclass of hierarchical networks, characterized by the absence of feedback loops and the normalization of inputs. We show that most optimal networks are directed and non-diagonalizable, necessitating the extension of the framework. We also show how oriented spanning trees can be used to explicitly and systematically construct optimal networks under network topological constraints. Our results may provide insights into the evolutionary origin of structures in complex networks for which synchronization plays a significant role.

Original languageEnglish (US)
Pages (from-to)77-89
Number of pages13
JournalPhysica D: Nonlinear Phenomena
Issue number1-2
StatePublished - Dec 2006


  • Complex networks
  • Optimization
  • Synchronization

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics

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