Trading interactions for topology in scale-free networks

Abstract

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, $γ' = (γ- μ)/(1-μ)$, describes how a shift of the standard exponent $γ$ of the degree distribution $P(q)$ can absorb the effect of degree-dependent pair interactions $J_{ij} \propto (q_iq_j)^{-μ}$. 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.