Stewart, Ian, 1945-. (2007) Elimination of multiple arrows and self-connections in coupled cell networks. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, 17 (1). pp. 99-106. ISSN 0218-1274

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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
Subjects:	Q Science > QA Mathematics Q Science
Journal or Publication Title:	INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
Publisher:	WORLD SCIENTIFIC PUBL CO PTE LTD
ISSN:	0218-1274
Date:	January 2007
Volume:	17
Number:	1
Number of Pages:	8
Page Range:	pp. 99-106
Identification Number:	10.1142/S0218127407017197
Publication Status:	Published
URI:	http://wrap.warwick.ac.uk/id/eprint/31164

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