Trading interactions for topology in scale-free networks

TL;DR

This paper investigates how degree-dependent interactions in scale-free networks influence their critical behavior, revealing a tunable universality class through a mapping that relates interaction exponents to topology.

Contribution

It introduces a mapping that relates the degree distribution exponent to interaction parameters, enabling exploration of universality classes by tuning interactions.

Findings

The universality class depends on both topology and interactions.

A mapping relates the degree distribution exponent to interaction parameters.

Supported by replica, cavity, and Monte Carlo methods.

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