title: Inter-layer synchronization in non-identical multi-layer networks.
abstract: Inter-layer synchronization is a dynamical process occurring in multi-layer networks composed of identical nodes. This process emerges when all layers are synchronized, while nodes in each layer do not necessarily evolve in unison. So far, the study of such inter-layer synchronization has been restricted to the case in which all layers have an identical connectivity structure. When layers are not identical, the inter-layer synchronous state is no longer a stable solution of the system. Nevertheless, when layers differ in just a few links, an approximate treatment is still feasible, and allows one to gather information on whether and how the system may wander around an inter-layer synchronous configuration. We report the details of an approximate analytical treatment for a two-layer multiplex, which results in the introduction of an extra inertial term accounting for structural differences. Numerical validation of the predictions highlights the usefulness of our approach, especially for small or moderate topological differences in the intra-layer coupling. Moreover, we identify a non-trivial relationship connecting the betweenness centrality of the missing links and the intra-layer coupling strength. Finally, by the use of multiplexed layers of electronic circuits, we study the inter-layer synchronization as a function of the removed links.
authors.affiliation: Complex Systems Group &GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain.|Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain.
authors.forename: I
authors.initials: I
authors.lastname: Leyva
authors.affiliation: Centro Universitario de los Lagos, Universidad de Guadalajara, Jalisco 47460, Mexico.
authors.forename: R
authors.initials: R
authors.lastname: Sevilla-Escoboza
authors.affiliation: Complex Systems Group &GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain.|Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain.
authors.forename: I
authors.initials: I
authors.lastname: Sendiña-Nadal
authors.affiliation: School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK.
authors.forename: R
authors.initials: R
authors.lastname: Gutiérrez
authors.affiliation: Complex Systems Group &GISC, Universidad Rey Juan Carlos, 28933 Móstoles, Madrid, Spain.|Center for Biomedical Technology, Universidad Politécnica de Madrid, 28223 Pozuelo de Alarcón, Madrid, Spain.
authors.forename: J M
authors.initials: JM
authors.lastname: Buldú
authors.affiliation: CNR-Institute of complex systems, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy.|The Italian Embassy in Israel, Hamered Street 25, 68125 Tel Aviv, Israel.
authors.forename: S
authors.initials: S
authors.lastname: Boccaletti