  We consider a recently introduced generalization of the Ising model in which
individual spin strength can vary. The model is intended for analysis of
ordering in systems comprising agents which, although matching in their
binarity (i.e., maintaining the iconic Ising features of `+' or `$-$', `up' or
`down', `yes' or `no'), differ in their strength. To investigate the interplay
between variable properties of nodes and interactions between them, we study
the model on a complex network where both the spin strength and degree
distributions are governed by power laws. We show that in the annealed network
approximation, thermodynamic functions of the model are self-averaging and we
obtain an exact solution for the partition function. This allows us to derive
the leading temperature and field dependencies of thermodynamic functions,
their critical behavior, and logarithmic corrections at the interface of
different phases. We find the delicate interplay of the two power laws leads to
new universality classes.
