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LIT-1674: Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks

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Properties5
StatusActive — read in full 2026-10-27 (NOTE-1217), from the accepted version in the Warwick repository; worth reading as the theorem behind robust synchrony in networks: an architecture forces a synchrony pattern for every admissible dynamics exactly when the pattern is balanced, which symmetry implies but does not exhaust, and the dynamics on it are a quotient network's. It treats existence only, not stability.
Version2
Tagscomplex-systems, network-science, mathematics
Cited as “paper” byNOTE-1217 — Stewart, Golubitsky & Pivato — Symmetry groupoids and patterns of synchrony
Cited as “source” byTHEORY-1059 — The dynamics a coupled cell network admits, restricted to a robustly synchronous pattern, are dynamics admitted by a smaller quotient network whose cells are the synchronized classes, so behaviour of the quotient, including symmetric behaviour the original network has no symmetry to force, appears in the original as synchronous behaviour; the quotient admits more dynamics than restriction produces, and is unique only up to refinement of its arrow types, THEORY-1065 — In a coupled cell network, a pattern of synchrony is invariant under every dynamics the network's architecture admits if and only if its partition of the cells is balanced, each cell of a class receiving, through each kind of arrow, the same number of inputs from each class; symmetry is one way to get balance and not the only one, so robust synchrony occurs in networks with no symmetry and among cells with no coupling between them, and the invariance says nothing about whether the pattern attracts

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Stewart, I., Golubitsky, M., & Pivato, M. (2003). Symmetry groupoids and patterns of synchrony in coupled cell networks. SIAM Journal on Applied Dynamical Systems, 2(4), 609–646. doi:10.1137/S1111111103419896.

Read in full on 2026-10-27 (NOTE-1217).

Why this is here

A22 §7 used it for the distinction between an invariant organization and an attracting one: network architecture fixes which synchrony patterns are possible, and dynamical stability which are realized. The record’s THEORY-276 is the symmetry-group case it generalizes. It bears on CLAIM-471 and QUESTION-063.

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