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Multiplex congruence network of natural numbers

Congruence theory has many applications in physical, social, biological and technological systems. Congruence arithmetic has been a fundamental tool for data security and computer algebra. However, much less attention was devoted to the topological features of congruence relations among natural numb...

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Autores principales: Yan, Xiao-Yong, Wang, Wen-Xu, Chen, Guan-Rong, Shi, Ding-Hua
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Nature Publishing Group 2016
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4814780/
https://www.ncbi.nlm.nih.gov/pubmed/27029650
http://dx.doi.org/10.1038/srep23714
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author Yan, Xiao-Yong
Wang, Wen-Xu
Chen, Guan-Rong
Shi, Ding-Hua
author_facet Yan, Xiao-Yong
Wang, Wen-Xu
Chen, Guan-Rong
Shi, Ding-Hua
author_sort Yan, Xiao-Yong
collection PubMed
description Congruence theory has many applications in physical, social, biological and technological systems. Congruence arithmetic has been a fundamental tool for data security and computer algebra. However, much less attention was devoted to the topological features of congruence relations among natural numbers. Here, we explore the congruence relations in the setting of a multiplex network and unveil some unique and outstanding properties of the multiplex congruence network. Analytical results show that every layer therein is a sparse and heterogeneous subnetwork with a scale-free topology. Counterintuitively, every layer has an extremely strong controllability in spite of its scale-free structure that is usually difficult to control. Another amazing feature is that the controllability is robust against targeted attacks to critical nodes but vulnerable to random failures, which also differs from ordinary scale-free networks. The multi-chain structure with a small number of chain roots arising from each layer accounts for the strong controllability and the abnormal feature. The multiplex congruence network offers a graphical solution to the simultaneous congruences problem, which may have implication in cryptography based on simultaneous congruences. Our work also gains insight into the design of networks integrating advantages of both heterogeneous and homogeneous networks without inheriting their limitations.
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spelling pubmed-48147802016-04-04 Multiplex congruence network of natural numbers Yan, Xiao-Yong Wang, Wen-Xu Chen, Guan-Rong Shi, Ding-Hua Sci Rep Article Congruence theory has many applications in physical, social, biological and technological systems. Congruence arithmetic has been a fundamental tool for data security and computer algebra. However, much less attention was devoted to the topological features of congruence relations among natural numbers. Here, we explore the congruence relations in the setting of a multiplex network and unveil some unique and outstanding properties of the multiplex congruence network. Analytical results show that every layer therein is a sparse and heterogeneous subnetwork with a scale-free topology. Counterintuitively, every layer has an extremely strong controllability in spite of its scale-free structure that is usually difficult to control. Another amazing feature is that the controllability is robust against targeted attacks to critical nodes but vulnerable to random failures, which also differs from ordinary scale-free networks. The multi-chain structure with a small number of chain roots arising from each layer accounts for the strong controllability and the abnormal feature. The multiplex congruence network offers a graphical solution to the simultaneous congruences problem, which may have implication in cryptography based on simultaneous congruences. Our work also gains insight into the design of networks integrating advantages of both heterogeneous and homogeneous networks without inheriting their limitations. Nature Publishing Group 2016-03-31 /pmc/articles/PMC4814780/ /pubmed/27029650 http://dx.doi.org/10.1038/srep23714 Text en Copyright © 2016, Macmillan Publishers Limited http://creativecommons.org/licenses/by/4.0/ This work is licensed under a Creative Commons Attribution 4.0 International License. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in the credit line; if the material is not included under the Creative Commons license, users will need to obtain permission from the license holder to reproduce the material. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/
spellingShingle Article
Yan, Xiao-Yong
Wang, Wen-Xu
Chen, Guan-Rong
Shi, Ding-Hua
Multiplex congruence network of natural numbers
title Multiplex congruence network of natural numbers
title_full Multiplex congruence network of natural numbers
title_fullStr Multiplex congruence network of natural numbers
title_full_unstemmed Multiplex congruence network of natural numbers
title_short Multiplex congruence network of natural numbers
title_sort multiplex congruence network of natural numbers
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4814780/
https://www.ncbi.nlm.nih.gov/pubmed/27029650
http://dx.doi.org/10.1038/srep23714
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