As pointed by , in the class of coupled cell networks that permits self-coupling and multiarrows, it is possible for two different coupled cell networks to generate the same space of admissible vector fields, i.e., to be ODE-equivalent.
In  it is shown that two coupled cell networks are ODE-equivalent if and only if they are linearly equivalent. Basically, the ODE-equivalence reduces to `linear equivalence', where two networks (with suitably identified phase spaces) are linearly equivalent if they determine the same space of linear admissible vector fields.
In this work we address the question: Given an ODE-equivalence class of coupled cell networks, is there a subclass of networks such that the number of edges is minimal among all the networks of that ODE-class?
 M. Golubitsky, I. Stewart and A. Torok., "Patterns of Synchrony in Coupled Cell Networks with Multiple Arrows", SIAM J. Appl. Dynam. Sys., to appear.
 A.P.S. Dias and I. Stewart. "Linear Equivalence and ODE-equivalence for Coupled Cell Networks", submitted.
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