A formal theory of symmetries of networks of coupled dynamical
systems, stated in terms of the group of permutations of the nodes that preserve
the network topology, has existed for some time. Global network symmetries
impose strong constraints on the corresponding dynamical systems,
which affect equilibria, periodic states, heteroclinic cycles, and even chaotic
states. In particular, the symmetries of the network can lead to synchrony,
phase relations, resonances, and synchronous or cycling chaos.
Symmetry is a rather restrictive assumption, and a general theory of networks
should be more flexible. A recent generalization of the group-theoretic
notion of symmetry replaces global symmetries by bijections between certain
subsets of the directed edges of the network, the ‘input sets’. Now the symmetry
group becomes a groupoid, which is an algebraic structure that resembles
a group, except that the product of two elements may not be defined. The
groupoid formalism makes it possible to extend group-theoretic methods to
more general networks, and in particular it leads to a complete classification
of ‘robust’ patterns of synchrony in terms of the combinatorial structure of the
network.
Many phenomena that would be nongeneric in an arbitrary dynamical
system can become generic when constrained by a particular network topology.
A network of dynamical systems is not just a dynamical system with
a high-dimensional phase space. It is also equipped with a canonical set of
observables—the states of the individual nodes of the network. Moreover, the
form of the underlying ODE is constrained by the network topology—which
variables occur in which component equations, and how those equations relate
to each other. The result is a rich and new range of phenomena, only a few of
which are yet properly understood
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