Bick Christian, Sclosa Davide
Department of Mathematics, Vrije Universiteit Amsterdam, 1081 HV Amsterdam, The Netherlands.
Institute for Advanced Study, Technische Universität München, 85748 Garching, Germany.
J Dyn Differ Equ. 2025;37(2):1171-1206. doi: 10.1007/s10884-023-10334-7. Epub 2024 Feb 23.
The collective dynamics of interacting dynamical units on a network crucially depends on the properties of the network structure. Rather than considering large but finite graphs to capture the network, one often resorts to graph limits and the dynamics thereon. We elucidate the symmetry properties of dynamical systems on graph limits-including graphons and graphops-and analyze how the symmetry shapes the dynamics, for example through invariant subspaces. In addition to traditional symmetries, dynamics on graph limits can support generalized noninvertible symmetries. Moreover, as asymmetric networks can have symmetric limits, we note that one can expect to see ghosts of symmetries in the dynamics of large but finite asymmetric networks.
网络上相互作用的动力学单元的集体动力学关键取决于网络结构的属性。与其考虑大型但有限的图来描述网络，人们通常求助于图极限及其上的动力学。我们阐明了图极限（包括图子和图运算）上动力学系统的对称性质，并分析了对称性如何塑造动力学，例如通过不变子空间。除了传统对称性外，图极限上的动力学还可以支持广义非可逆对称性。此外，由于不对称网络可以有对称极限，我们注意到在大型但有限的不对称网络的动力学中可以预期看到对称性的“幽灵”。
