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Elimination of multiple arrows and self-connections in coupled cell networks
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Stewart, Ian (2007) Elimination of multiple arrows and self-connections in coupled cell networks. International Journal of Bifurcation and Chaos, Volume 17 (Number 1). pp. 99-106. doi:10.1142/S0218127407017197 ISSN 0218-1274.
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Official URL: http://dx.doi.org/10.1142/S0218127407017197
Abstract
A coupled cell network is a finite directed graph in which nodes and edges are classified into equivalence classes. Such networks arise in a formal theory of coupled systems of differential equations, as a schematic indication of the topology of the coupling, but they can be studied independently as combinatorial objects. The edges of a coupled cell network are "identical" if they are all equivalent, and the network is "homogeneous" if all nodes have isomorphic sets of input edges. Golubitsky et al. [ 2005] proved that every homogeneous identical-edge coupled cell network is a quotient of a network that has no multiple edges and no self-connections. We generalize this theorem to any coupled cell network by removing the conditions of homogeneity and identical edges. The problem is a purely combinatorial assertion about labeled directed graphs, and we give two combinatorial proofs. Both proofs eliminate self-connections inductively. The first proof also eliminates multiple edges inductively, the main feature being the specification of the inductive step in terms of a complexity measure. The second proof obtains a more efficient result by eliminating all multiple edges in a single construction.
Item Type: | Journal Article | ||||
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Subjects: | Q Science > QA Mathematics Q Science |
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | International Journal of Bifurcation and Chaos | ||||
Publisher: | World Scientific Publishing Co. Pte. Ltd. | ||||
ISSN: | 0218-1274 | ||||
Official Date: | January 2007 | ||||
Dates: |
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Volume: | Volume 17 | ||||
Number: | Number 1 | ||||
Number of Pages: | 8 | ||||
Page Range: | pp. 99-106 | ||||
DOI: | 10.1142/S0218127407017197 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Open Access (Creative Commons) |
Data sourced from Thomson Reuters' Web of Knowledge
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