Classification of critical phenomena in hierarchical small-world networks

TL;DR

This paper classifies critical phenomena in hierarchical small-world networks, revealing three regimes of phase transitions with unique characteristics, and explains the occurrence of inverted BKT transitions in complex networks.

Contribution

It introduces a new classification scheme for critical behavior in control-parameter dependent systems, applicable to hierarchical networks and related models.

Findings

Identifies three distinct critical regimes in hierarchical networks.

Provides a local analysis using cubic recursion equations for RG flow.

Explains the prevalence of inverted BKT transitions in complex networks.

Abstract

A classification of critical behavior is provided in systems for which the renormalization group equations are control-parameter dependent. It describes phase transitions in networks with a recursive, hierarchical structure but appears to apply also to a wider class of systems, such as conformal field theories. Although these transitions generally do not exhibit universality, three distinct regimes of characteristic critical behavior can be discerned that combine an unusual mixture of finite- and infinite-order transitions. In the spirit of Landau's description of a phase transition, the problem can be reduced to the local analysis of a cubic recursion equation, here, for the renormalization group flow of some generalized coupling. Among other insights, this theory explains the often-noted prevalence of the so-called inverted Berezinskii-Kosterlitz-Thouless transitions in complex networks. As a demonstration, a one-parameter family of Ising models on hierarchical networks is considered.