  We exploit the symmetry concepts developed in the companion review of this
article to introduce a stochastic version of link reversal symmetry, which
leads to an improved understanding of the reciprocity of directed networks. We
apply our formalism to the international trade network and show that a strong
embedding in economic space determines particular symmetries of the network,
while the observed evolution of reciprocity is consistent with a symmetry
breaking taking place in production space. Our results show that networks can
be strongly affected by symmetry-breaking phenomena occurring in embedding
spaces, and that stochastic network symmetries can successfully suggest, or
rule out, possible underlying mechanisms.
