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Complex networks exhibit a wide range of collective dynamic phenomena, including synchronization, diffusion, relaxation, and coordination processes. Their asymptotic dynamics is generically characterized by the local Jacobian, graph Laplacian, or a similar linear operator. The structure of networks with regular, small-world, and random connectivities are reasonably well understood, but their collective dynamical properties remain largely unknown. Here we present a two-stage mean-field theory to derive analytic expressions for network spectra. A single formula covers the spectrum from regular via small-world to strongly randomized topologies in Watts-Strogatz networks, explaining the simultaneous dependencies on network size N, average degree k, and topological randomness q. We present simplified analytic predictions for the second-largest and smallest eigenvalue, and numerical checks confirm our theoretical predictions for zero, small, and moderate topological randomness q, including the entire small-world regime. For large q of the order of one, we apply standard random matrix theory, thereby overarching the full range from regular to randomized network topologies. These results may contribute to our analytic and mechanistic understanding of collective relaxation phenomena of network dynamical systems.

Original languageEnglish
Article number052815
Number of pages11
JournalPhysical Review. E, Statistical, Nonlinear and Soft Matter Physics
Issue number5
Publication statusPublished - 27 May 2015

Bibliographical note

This work was supported by the BMBF, Grants No. 03SF0472A (C.G., J.K.) and No. 03SF0472E (M.T.), by a grant of the Max Planck Society (M.T.), and by the Engineering and Physical Sciences Research Council (EPSRC) Grant No. EP/E501311/1 (S.G.).


  • sparse random matrices
  • complex networks
  • circular law
  • synchronization
  • systems
  • laplacian
  • spectrum
  • model
  • localization
  • universality


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