  An anomalous mean-field solution is known to capture the non trivial phase
diagram of the Ising model in annealed complex networks. Nevertheless the
critical fluctuations in random complex networks remain mean-field. Here we
show that a break-down of this scenario can be obtained when complex networks
are embedded in geometrical spaces. Through the analysis of the Ising model on
annealed spatial networks, we reveal in particular the spectral properties of
networks responsible for critical fluctuations and we generalize the Ginsburg
criterion to complex topologies.
