  We consider spin models on complex networks frequently used to model social
and technological systems. We study the annealed ferromagnetic Ising model for
random networks with either independent edges (Erd\H{o}s-R\'enyi), or with
prescribed degree distributions (configuration model). Contrary to many
physical models, the annealed setting is poorly understood and behaves quite
differently than the quenched system. In annealed networks with a fluctuating
number of edges, the Ising model changes the degree distribution, an aspect
previously ignored. For random networks with Poissonian degrees, this gives
rise to three distinct annealed critical temperatures depending on the precise
model choice, only one of which reproduces the quenched one. In particular, two
of these annealed critical temperatures are finite even when the quenched one
is infinite, since then the annealed graph creates a giant component for all
sufficiently small temperatures. We see that the critical exponents in the
configuration model with deterministic degrees are the same as the quenched
ones, which are the mean-field exponents if the degree distribution has finite
fourth moment, and power-law-dependent critical exponents otherwise.
Remarkably, the annealing for the configuration model with random i.i.d.
degrees washes away the universality class with power-law critical exponents.
