  We study the phase transition of the Ising model in networks with
core-periphery structures. By Monte Carlo simulations, we show that prior to
the order-disorder phase transition the system organizes into an inhomogeneous
intermediate phase in which core nodes are much more ordered than peripheral
nodes. Interestingly, the susceptibility shows double peaks at two distinct
temperatures. We find that, if the connections between core and periphery
increase linearly with network size, the first peak does not exhibit any
size-dependent effect, and the second one diverges in the limit of infinite
network size. Otherwise, if the connections between core and periphery scale
sub-linearly with the network size, both peaks of the susceptibility diverge as
power laws in the thermodynamic limit. This suggests the appearance of a double
transition phenomenon in the Ising model for the latter case. Moreover, we
develop a mean-field theory that agrees well with the simulations.
