  In this work we consider a superradiant phase transition problem for the
Dicke-Ising model, which generalizes the Dicke and Ising models for annealed
complex networks presuming spin-spin interaction. The model accounts the
interaction between a spin (two-level) system and external classical (magnetic)
and quantized (transverse) fields. We examine regular, random, and scale-free
network structures characterized by the delta-function, random (Poisson), and
power-law exponent degree distributions, respectively. To describe paramagnetic
(PM) - ferromagnetic (FM) and superradiant (SR) phase transitions we introduce
two order parameters: the total weighted spin z-component and the normalized
transverse field amplitude, which correspond to the spontaneous magnetization
in z and x directions, respectively. Due to the interplay between the spin
interaction and the finite size effects in the networks we first elucidate
novel features of the SR state in the presence of the PM-FM phase transition.
We reveal that the critical temperature of the SR phase transition grows
monotonically from some certain value that corresponds to the critical number
of nodes. For the scale-free networks we find that this critical temperature
rises with the degree exponent increase accompanied both by establishing
spontaneous magnetization in a quantum transverse field and vanishing of the
collective spin component in z direction. In addition, we establish the
conditions for the network parameters to obtain a quantum phase transition in
the spin system when the critical temperature approaches zero. The fundamental
features, which involve critical values of the classical and quantum field
parameters, are discussed for the occurrence of the superradiance phase in this
limit.
