Mean field limits of co-evolutionary signed heterogeneous networks
Many science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to...
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Cambridge University Press
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| Series: | European Journal of Applied Mathematics |
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| author | Marios Antonios Gkogkas Christian Kuehn Chuang Xu |
| author_facet | Marios Antonios Gkogkas Christian Kuehn Chuang Xu |
| author_sort | Marios Antonios Gkogkas |
| collection | DOAJ |
| description | Many science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to model myriad real-world phenomena as co-evolutionary (or adaptive) networks. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramoto-type networks have been widely used to model interaction among a set of oscillators. In this paper, we propose a rigorous formulation for limits of a sequence of co-evolutionary Kuramoto oscillators coupled on heterogeneous co-evolutionary networks, which receive both positive and negative feedback from the dynamics of the oscillators on the networks. We show under mild conditions, the mean field limit (MFL) of the co-evolutionary network exists and the sequence of co-evolutionary Kuramoto networks converges to this MFL. Such MFL is described by solutions of a generalised Vlasov equation. We treat the graph limits as signed graph measures, motivated by the recent work in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261–349]. In comparison to the recently emerging works on MFLs of IPS coupled on non-co-evolutionary networks (i.e., static networks or time-dependent networks independent of the dynamics of the IPS), our work seems the first to rigorously address the MFL of a co-evolutionary network model. The approach is based on our formulation of a generalisation of the co-evolutionary network as a hybrid system of ODEs and measure differential equations parametrised by a vertex variable, together with an analogue of the variation of parameters formula, as well as the generalised Neunzert’s in-cell-particle method developed in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261–349]. |
| format | Article |
| id | doaj-art-64ce8ca515ec4d37b261b81fec230a72 |
| institution | Kabale University |
| issn | 0956-7925 1469-4425 |
| language | English |
| publisher | Cambridge University Press |
| record_format | Article |
| series | European Journal of Applied Mathematics |
| spelling | doaj-art-64ce8ca515ec4d37b261b81fec230a722025-01-16T21:52:44ZengCambridge University PressEuropean Journal of Applied Mathematics0956-79251469-442514410.1017/S0956792524000858Mean field limits of co-evolutionary signed heterogeneous networksMarios Antonios Gkogkas0Christian Kuehn1Chuang Xu2https://orcid.org/0000-0002-8965-6043Department of Mathematics, Technical University of Munich, Munich, GermanyDepartment of Mathematics, Technical University of Munich, Munich, Germany Munich Data Science Institute (MDSI), Technical University of Munich, Munich, GermanyDepartment of Mathematics, Technical University of Munich, Munich, Germany Department of Mathematics, University of Hawai’i at Mānoa, Honolulu, Hawai’i, USAMany science phenomena are modelled as interacting particle systems (IPS) coupled on static networks. In reality, network connections are far more dynamic. Connections among individuals receive feedback from nearby individuals and make changes to better adapt to the world. Hence, it is reasonable to model myriad real-world phenomena as co-evolutionary (or adaptive) networks. These networks are used in different areas including telecommunication, neuroscience, computer science, biochemistry, social science, as well as physics, where Kuramoto-type networks have been widely used to model interaction among a set of oscillators. In this paper, we propose a rigorous formulation for limits of a sequence of co-evolutionary Kuramoto oscillators coupled on heterogeneous co-evolutionary networks, which receive both positive and negative feedback from the dynamics of the oscillators on the networks. We show under mild conditions, the mean field limit (MFL) of the co-evolutionary network exists and the sequence of co-evolutionary Kuramoto networks converges to this MFL. Such MFL is described by solutions of a generalised Vlasov equation. We treat the graph limits as signed graph measures, motivated by the recent work in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261–349]. In comparison to the recently emerging works on MFLs of IPS coupled on non-co-evolutionary networks (i.e., static networks or time-dependent networks independent of the dynamics of the IPS), our work seems the first to rigorously address the MFL of a co-evolutionary network model. The approach is based on our formulation of a generalisation of the co-evolutionary network as a hybrid system of ODEs and measure differential equations parametrised by a vertex variable, together with an analogue of the variation of parameters formula, as well as the generalised Neunzert’s in-cell-particle method developed in [Kuehn, Xu. Vlasov equations on digraph measures, JDE, 339 (2022), 261–349].https://www.cambridge.org/core/product/identifier/S0956792524000858/type/journal_articleadaptive networkssparse networksevolution equationssigned graph limitsgeneralised Vlasov equationKuramoto networks35R0292C4260B10 |
| spellingShingle | Marios Antonios Gkogkas Christian Kuehn Chuang Xu Mean field limits of co-evolutionary signed heterogeneous networks European Journal of Applied Mathematics adaptive networks sparse networks evolution equations signed graph limits generalised Vlasov equation Kuramoto networks 35R02 92C42 60B10 |
| title | Mean field limits of co-evolutionary signed heterogeneous networks |
| title_full | Mean field limits of co-evolutionary signed heterogeneous networks |
| title_fullStr | Mean field limits of co-evolutionary signed heterogeneous networks |
| title_full_unstemmed | Mean field limits of co-evolutionary signed heterogeneous networks |
| title_short | Mean field limits of co-evolutionary signed heterogeneous networks |
| title_sort | mean field limits of co evolutionary signed heterogeneous networks |
| topic | adaptive networks sparse networks evolution equations signed graph limits generalised Vlasov equation Kuramoto networks 35R02 92C42 60B10 |
| url | https://www.cambridge.org/core/product/identifier/S0956792524000858/type/journal_article |
| work_keys_str_mv | AT mariosantoniosgkogkas meanfieldlimitsofcoevolutionarysignedheterogeneousnetworks AT christiankuehn meanfieldlimitsofcoevolutionarysignedheterogeneousnetworks AT chuangxu meanfieldlimitsofcoevolutionarysignedheterogeneousnetworks |