Criticality on networks with topology-dependent interactions

TL;DR

This paper investigates how the critical behavior of networks with topology-dependent interactions can be tuned by modifying interaction forms, revealing a universal mapping that relates degree distribution exponents to interaction parameters.

Contribution

It introduces a universal mapping that relates degree distribution exponents to interaction parameters, applicable to both equilibrium and nonequilibrium network models.

Findings

The mapping $ ext{γ'}=( ext{γ}- ext{μ})/(1- ext{μ})$ describes how degree distribution exponents shift with interaction form.

Numerical simulations confirm the theoretical prediction across different models.

Critical temperature estimates are obtained using Bethe-Peierls and replica methods.

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 $\gamma'=(\gamma-\mu)/(1-\mu)$ 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.