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Characterization of fundamental networks

Part of: Graph theory

Published online by Cambridge University Press:  26 January 2019

Manuela A. D. Aguiar ,
Ana P. S. Dias  and
Pedro Soares
Manuela A. D. Aguiar
Affiliation:
Faculdade de Economia, Universidade do Porto, Rua Dr Roberto Frias, 4200-464 Porto, Portugal and Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007Porto, Portugal ([email protected])
Ana P. S. Dias
Affiliation:
Dep. Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007Porto, Portugal and Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal ([email protected]; [email protected])
Pedro Soares
Affiliation:
Dep. Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre, 687, 4169-007Porto, Portugal and Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal ([email protected]; [email protected])
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Abstract

In the framework of coupled cell systems, a coupled cell network describes graphically the dynamical dependencies between individual dynamical systems, the cells. The fundamental network of a network reveals the hidden symmetries of that network. Subspaces defined by equalities of coordinates which are flow-invariant for any coupled cell system consistent with a network structure are called the network synchrony subspaces. Moreover, for every synchrony subspace, each network admissible system restricted to that subspace is a dynamical system consistent with a smaller network called a quotient network. We characterize networks such that: the network is a subnetwork of its fundamental network, and the network is a fundamental network. Moreover, we prove that the fundamental network construction preserves the quotient relation and it transforms the subnetwork relation into the quotient relation. The size of cycles in a network and the distance of a cell to a cycle are two important properties concerning the description of the network architecture. In this paper, we relate these two architectural properties in a network and its fundamental network.

Keywords

Fundamental networksnetwork connectivityrings

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary: 05C60: Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 05C40: Connectivity
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 453 - 474
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Aguiar, M., Dias, A., Golubitsky, M. and Leite, M.. Bifurcations from regular quotient networks: a first insight. Phys. D 238 (2009), 137–155.CrossRefGoogle Scholar
2Biggs, N.. Algebraic graph theory. Cambridge tracts in Mathematics, vol. 67 (London: Cambridge University Press, 1974).CrossRefGoogle Scholar
3Boldi, P. and Vigna, S.. Fibrations of graphs. Discrete Math. 243 (2002), 21–66.CrossRefGoogle Scholar
4DeVille, L. and Lerman, E.. Modular dynamical systems on networks. J. Eur. Math. Soc. (JEMS) 17 (2015), 2977–3013.CrossRefGoogle Scholar
5Ganbat, A.. Reducibility of steady-state bifurcations in coupled cell systems. J. Math. Anal. Appl. 415 (2014), 159–177.CrossRefGoogle Scholar
6Golubitsky, M., Stewart, I. and Török, A.. Patterns of synchrony in coupled cell networks with multiple arrows. SIAM J. Appl. Dyn. Syst. 4 (2005), 78–100.CrossRefGoogle Scholar
7Kamei, H.. The existence and classification of synchrony-breaking bifurcations in regular homogeneous networks using lattice structures. Int. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), 3707–3732.CrossRefGoogle Scholar
8Moreira, C.. On bifurcations in lifts of regular uniform coupled cell networks. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2014), 20140241.CrossRefGoogle Scholar
9Nijholt, E., Rink, B. and Sanders, J.. Graph fibrations and symmetries of network dynamics. J. Differ. Equ. 261 (2016a), 4861–4896.CrossRefGoogle Scholar
10Nijholt, E., Rink, B. and Sanders, J., Projection blocks in homogeneous coupled cell networks, arXiv preprint arXiv:1612.05213 (2016b).CrossRefGoogle Scholar
11Rink, B. and Sanders, J.. Coupled cell networks and their hidden symmetries. SIAM J. Math. Anal. 46 (2014), 1577–1609.CrossRefGoogle Scholar
12Rink, B. and Sanders, J.. Coupled cell networks: semigroups, Lie algebras and normal forms. Trans. Amer. Math. Soc. 367 (2015), 3509–3548.CrossRefGoogle Scholar
13Stewart, I., Golubitsky, M. and Pivato, M.. Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM J. Appl. Dyn. Syst. 2 (2003), 609–646.CrossRefGoogle Scholar