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A New Decomposition of the Graph Laplacian and the Binomial Structure of Mass-Action Systems

We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible core matrix, the vector of tree constants, and the incidence matrix of an auxiliary graph, representing an order on the vertices. Depending on the particul...

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Autor principal: Müller, Stefan
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2023
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10397138/
https://www.ncbi.nlm.nih.gov/pubmed/37546229
http://dx.doi.org/10.1007/s00332-023-09942-w
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author Müller, Stefan
author_facet Müller, Stefan
author_sort Müller, Stefan
collection PubMed
description We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible core matrix, the vector of tree constants, and the incidence matrix of an auxiliary graph, representing an order on the vertices. Depending on the particular order, the core matrix has additional properties. Our results are graph-theoretic/algebraic in nature. As a first application, we further clarify the binomial structure of (weakly reversible) mass-action systems, arising from chemical reaction networks. Second, we extend a classical result by Horn and Jackson on the asymptotic stability of special steady states (complex-balanced equilibria). Here, the new decomposition of the graph Laplacian allows us to consider regions in the positive orthant with given monomial evaluation orders (and corresponding polyhedral cones in logarithmic coordinates). As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain binomial differential inclusions. In particular, this holds for complex-balanced mass-action systems, and hence, we also obtain a polyhedral-geometry proof of the classical result.
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spelling pubmed-103971382023-08-04 A New Decomposition of the Graph Laplacian and the Binomial Structure of Mass-Action Systems Müller, Stefan J Nonlinear Sci Article We provide a new decomposition of the Laplacian matrix (for labeled directed graphs with strongly connected components), involving an invertible core matrix, the vector of tree constants, and the incidence matrix of an auxiliary graph, representing an order on the vertices. Depending on the particular order, the core matrix has additional properties. Our results are graph-theoretic/algebraic in nature. As a first application, we further clarify the binomial structure of (weakly reversible) mass-action systems, arising from chemical reaction networks. Second, we extend a classical result by Horn and Jackson on the asymptotic stability of special steady states (complex-balanced equilibria). Here, the new decomposition of the graph Laplacian allows us to consider regions in the positive orthant with given monomial evaluation orders (and corresponding polyhedral cones in logarithmic coordinates). As it turns out, all dynamical systems are asymptotically stable that can be embedded in certain binomial differential inclusions. In particular, this holds for complex-balanced mass-action systems, and hence, we also obtain a polyhedral-geometry proof of the classical result. Springer US 2023-08-02 2023 /pmc/articles/PMC10397138/ /pubmed/37546229 http://dx.doi.org/10.1007/s00332-023-09942-w Text en © The Author(s) 2023 https://creativecommons.org/licenses/by/4.0/Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/ (https://creativecommons.org/licenses/by/4.0/) .
spellingShingle Article
Müller, Stefan
A New Decomposition of the Graph Laplacian and the Binomial Structure of Mass-Action Systems
title A New Decomposition of the Graph Laplacian and the Binomial Structure of Mass-Action Systems
title_full A New Decomposition of the Graph Laplacian and the Binomial Structure of Mass-Action Systems
title_fullStr A New Decomposition of the Graph Laplacian and the Binomial Structure of Mass-Action Systems
title_full_unstemmed A New Decomposition of the Graph Laplacian and the Binomial Structure of Mass-Action Systems
title_short A New Decomposition of the Graph Laplacian and the Binomial Structure of Mass-Action Systems
title_sort new decomposition of the graph laplacian and the binomial structure of mass-action systems
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC10397138/
https://www.ncbi.nlm.nih.gov/pubmed/37546229
http://dx.doi.org/10.1007/s00332-023-09942-w
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