Network synchronization, diffusion, and the paradox of heterogeneity

Adilson E. Motter*, Changsong ZHOU, Jürgen Kurths

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

466 Citations (Scopus)


Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states. We show that, for a given degree distribution, the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum. This enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size. Numerical verification of the main results is provided for representative classes of small-world and scale-free networks.

Original languageEnglish
Article number016116
JournalPhysical Review E
Issue number1
Publication statusPublished - Jan 2005

Scopus Subject Areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


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