  Weighted scale-free networks with topology-dependent interactions are
studied. It is shown that the possible universality classes of critical
behaviour, which are known to depend on topology, can also be explored by
tuning the form of the interactions at fixed topology. For a model of opinion
formation, simple mean field and scaling arguments show that a mapping
$\gamma'=(\gamma-\mu)/(1-\mu)$ describes how a shift of the standard exponent
$\gamma$ of the degree distribution can absorb the effect of degree-dependent
pair interactions $J_{ij} \propto (k_ik_j)^{-\mu}$, where $k_i$ stands for the
degree of vertex $i$. This prediction is verified by extensive numerical
investigations using the cavity method and Monte Carlo simulations. The
critical temperature of the model is obtained through the Bethe-Peierls
approximation and with the replica technique. The mapping can be extended to
nonequilibrium models such as those describing the spreading of a disease on a
network.
