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Table 1 Some permutation symmetry groups that have been considered as examples of symmetries of coupled oscillator networks

From: Mathematical Frameworks for Oscillatory Network Dynamics in Neuroscience

Name Symbol Comments
Full permutation \(S_{N}\) Global or all-to-all coupling [118, 120]
Undirected ring \(\mathbb {D}_{N}\) Dihedral symmetry [118, 120]
Directed ring \(\mathbb {Z}_{N}\) Cyclic symmetry [118, 120]
Polyhedral networks Various [121]
Lattice networks \(G_{1}\times G_{2}\) \(G_{1}\) and \(G_{2}\) could be \(\mathbb {D}_{k}\) or \(\mathbb {Z}_{k}\)
Hierarchical networks \(G_{1} \wr G_{2}\) \(G_{1}\) is the local symmetry, \(G_{2}\) the global symmetry, and is the wreath product [122]