  Scale-free networks with topology-dependent interactions are studied. It is
shown that the universality classes of critical behavior, which conventionally
depend only on topology, can also be explored by tuning the interactions. A
mapping, $\gamma' = (\gamma - \mu)/(1-\mu)$, describes how a shift of the
standard exponent $\gamma$ of the degree distribution $P(q)$ can absorb the
effect of degree-dependent pair interactions $J_{ij} \propto (q_iq_j)^{-\mu}$.
Replica technique, cavity method and Monte Carlo simulation support the
physical picture suggested by Landau theory for the critical exponents and by
the Bethe-Peierls approximation for the critical temperature. The equivalence
of topology and interaction holds for equilibrium and non-equilibrium systems,
and is illustrated with interdisciplinary applications.
