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Graph partitions and cluster synchronization in networks of oscillators

Schaub, Michael T. and O'Clery, Neave and Billeh, Yazan N. and Delvenne, Jean-Charles and Lambiotte, Renaud and Barahona, Mauricio (2016) Graph partitions and cluster synchronization in networks of oscillators. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26 (9). Art. No. 094821. ISSN 1054-1500. PMCID PMC5381716. https://resolver.caltech.edu/CaltechAUTHORS:20160823-110841946

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Abstract

Synchronization over networks depends strongly on the structure of the coupling between the oscillators. When the coupling presents certain regularities, the dynamics can be coarse-grained into clusters by means of External Equitable Partitions of the network graph and their associated quotient graphs. We exploit this graph-theoretical concept to study the phenomenon of cluster synchronization, in which different groups of nodes converge to distinct behaviors. We derive conditions and properties of networks in which such clustered behavior emerges and show that the ensuing dynamics is the result of the localization of the eigenvectors of the associated graph Laplacians linked to the existence of invariant subspaces. The framework is applied to both linear and non-linear models, first for the standard case of networks with positive edges, before being generalized to the case of signed networks with both positive and negative interactions. We illustrate our results with examples of both signed and unsigned graphs for consensus dynamics and for partial synchronization of oscillator networks under the master stability function as well as Kuramoto oscillators.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1063/1.4961065DOIArticle
http://scitation.aip.org/content/aip/journal/chaos/26/9/10.1063/1.4961065PublisherArticle
https://arxiv.org/abs/1608.04283arXivDiscussion Paper
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5381716/PubMed CentralArticle
ORCID:
AuthorORCID
Schaub, Michael T.0000-0003-2426-6404
Billeh, Yazan N.0000-0001-5200-4992
Additional Information:© 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). Received 26 March 2016; accepted 1 August 2016; published online 19 August 2016. We thank Karol Bacik for comments and carefully reading the manuscript. MTS, JCD, RL acknowledge support from FRS-FNRS; the Belgian Network DYSCO (Dynamical Systems, Control, and Optimisation) funded by the Interuniversity Attraction Poles Programme initiated by the Belgian State, Science Policy Office; and the ARC (Action de Recherche Concerte) on Mining and Optimization of Big Data Models funded by the Wallonia-Brussels Federation. NO'C was funded by a Wellcome Trust Doctoral Studentship at Imperial College London during this work. YNB thanks the G. Harold and Leila Y. Mathers Foundation. MB acknowledges support through EPSRC grants EP/I017267/1 and EP/N014529/1. Data Availability: No new data was collected in the course of this research.
Funders:
Funding AgencyGrant Number
Belgian Federal Science Policy Office (BELSPO)UNSPECIFIED
Wallonia-Brussels FederationUNSPECIFIED
Wellcome TrustUNSPECIFIED
G. Harold and Leila Y. Mathers FoundationUNSPECIFIED
Engineering and Physical Sciences Research Council (EPSRC)EP/I017267/1
Engineering and Physical Sciences Research Council (EPSRC)EP/N014529/1
Fonds de la Recherche Scientifique (FNRS)UNSPECIFIED
Issue or Number:9
PubMed Central ID:PMC5381716
Record Number:CaltechAUTHORS:20160823-110841946
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20160823-110841946
Official Citation:Graph partitions and cluster synchronization in networks of oscillators Schaub, Michael T. and O'Clery, Neave and Billeh, Yazan N. and Delvenne, Jean-Charles and Lambiotte, Renaud and Barahona, Mauricio, Chaos, 26, 094821 (2016), DOI:http://dx.doi.org/10.1063/1.4961065
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:69850
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:23 Aug 2016 18:21
Last Modified:03 Oct 2019 10:25

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