title: Collective relaxation dynamics of small-world networks.
abstract: 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.
authors.affiliation: Research Domain on Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.|Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany.
authors.forename: Carsten
authors.initials: C
authors.lastname: Grabow
authors.affiliation: Mathematics Institute and Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, United Kingdom.
authors.forename: Stefan
authors.initials: S
authors.lastname: Grosskinsky
authors.affiliation: Research Domain on Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, P.O. Box 60 12 03, 14412 Potsdam, Germany.|Department of Physics, Humboldt University of Berlin, Newtonstr. 15, 12489 Berlin, Germany.|Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom.
authors.forename: Jürgen
authors.initials: J
authors.lastname: Kurths
authors.affiliation: Network Dynamics, Max Planck Institute for Dynamics and Self-Organization (MPIDS), 37077 Göttingen, Germany.|Institute for Nonlinear Dynamics, Faculty for Physics, Georg August University Göttingen, 37077 Göttingen, Germany.|Bernstein Center for Computational Neuroscience Göttingen, 37077 Göttingen, Germany.
authors.forename: Marc
authors.initials: M
authors.lastname: Timme
mesh_terms: D004058:Diffusion; D008962:Models, Theoretical; D013269:Stochastic Processes