Geometric unfolding of synchronization dynamics on networks
- 1 June 2021
- journal article
- research article
- Published by AIP Publishing in Chaos: An Interdisciplinary Journal of Nonlinear Science
- Vol. 31 (6), 061105
- https://doi.org/10.1063/5.0053837
Abstract
We study the synchronized state in a population of network-coupled, heterogeneous oscillators. In particular, we show that the steady-state solution of the linearized dynamics may be written as a geometric series whose subsequent terms represent different spatial scales of the network. Specifically, each additional term incorporates contributions from wider network neighborhoods. We prove that this geometric expansion converges for arbitrary frequency distributions and for both undirected and directed networks provided that the adjacency matrix is primitive. We also show that the error in the truncated series grows geometrically with the second largest eigenvalue of the normalized adjacency matrix, analogously to the rate of convergence to the stationary distribution of a random walk. Last, we derive a local approximation for the synchronized state by truncating the spatial series, at the first neighborhood term, to illustrate the practical advantages of our approach.Keywords
Funding Information
- Ministerio de Economía, Industria y Competitividad, Gobierno de España (PGC2018-094754-B-C2)
- Generalitat de Catalunya (Grant No. 2017SGR-896)
- Universitat Rovira i Virgili (Grant No. 2017PFR-URV-B2-41)
- Institució Catalana de Recerca i Estudis Avançats
- James S. McDonnell Foundation (Grant No. 220020325)
This publication has 34 references indexed in Scilit:
- Braess's paradox in oscillator networks, desynchronization and power outageNew Journal of Physics, 2012
- A sensing array of radically coupled genetic ‘biopixels’Nature, 2011
- Synchronization in complex networksPhysics Reports, 2008
- Communicability in complex networksPhysical Review E, 2008
- Analysis of nonlinear synchronization dynamics of oscillator networks by Laplacian spectral methodsPhysical Review E, 2007
- Synchronization is optimal in nondiagonalizable networksPhysical Review E, 2006
- Synchronization Reveals Topological Scales in Complex NetworksPhysical Review Letters, 2006
- A measure of betweenness centrality based on random walksSocial Networks, 2005
- Growing scale-free networks with tunable clusteringPhysical Review E, 2002
- Master Stability Functions for Synchronized Coupled SystemsPhysical Review Letters, 1998