Elimination of multiple arrows and self-connections in coupled cell networks
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. ISSN 0218-1274Full text not available from this repository.
Official URL: http://dx.doi.org/10.1142/S0218127407017197
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
|Divisions:||Faculty of Science > Mathematics|
|Journal or Publication Title:||International Journal of Bifurcation and Chaos|
|Publisher:||World Scientific Publishing Co. Pte. Ltd.|
|Official Date:||January 2007|
|Number of Pages:||8|
|Page Range:||pp. 99-106|
|Access rights to Published version:||Open Access|
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