Criticality on networks with topology-dependent interactions

Abstract

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 ${\gamma }^{\prime }=(\gamma -\mu )/(1-\mu )$ describes how a shift of the standard exponent $\gamma$ of the degree distribution can absorb the effect of degree-dependent pair interactions $J_{ij}\propto (k_i{k}_j{)}^{-\mu }$, 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.

20 pages, 14 figures

Cited by in corpus (7)

• Critical phenomena in complex networks
• How to make a fragile network robust and vice versa
• BKT-like transition in the Potts model on an inhomogeneous annealed network
• Griffiths singularities and algebraic order in the exact solution of an Ising model on a fractal modular network
• Ising model on the Apollonian network with node dependent interactions
• Localization properties of a tight-binding electronic model on the Apollonian network
• Spin models on random graphs with controlled topologies beyond degree constraints