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.
| Original language | English |
|---|---|
| Article number | 052815 |
| Number of pages | 11 |
| Journal | Physical Review. E, Statistical, Nonlinear and Soft Matter Physics |
| Volume | 91 |
| Issue number | 5 |
| DOIs | |
| Publication status | Published - 27 May 2015 |
Keywords
- sparse random matrices
- complex networks
- circular law
- synchronization
- systems
- laplacian
- spectrum
- model
- localization
- universality
Cite this
Collective relaxation dynamics of small-world networks. / Grabow, Carsten; Grosskinsky, Stefan; Kurths, Juergen; Timme, Marc.
In: Physical Review. E, Statistical, Nonlinear and Soft Matter Physics, Vol. 91, No. 5, 052815, 27.05.2015.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Collective relaxation dynamics of small-world networks
AU - Grabow, Carsten
AU - Grosskinsky, Stefan
AU - Kurths, Juergen
AU - Timme, Marc
N1 - ACKNOWLEDGMENTS 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.).
PY - 2015/5/27
Y1 - 2015/5/27
N2 - 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.
AB - 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.
KW - sparse random matrices
KW - complex networks
KW - circular law
KW - synchronization
KW - systems
KW - laplacian
KW - spectrum
KW - model
KW - localization
KW - universality
U2 - 10.1103/PhysRevE.91.052815
DO - 10.1103/PhysRevE.91.052815
M3 - Article
VL - 91
JO - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics
JF - Physical Review. E, Statistical, Nonlinear and Soft Matter Physics
SN - 1539-3755
IS - 5
M1 - 052815
ER -