Classification of critical phenomena in hierarchical small-world 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.

6 pages, RevTex4, see http://www.physics.emory.edu/faculty/boettcher/ for more information

References in corpus (12)

• Critical phenomena in complex networks
• Ordinary Percolation with Discontinuous Transitions
• Small-World Bonds and Patchy Percolation on the Hanoi Network
• Griffiths singularities and algebraic order in the exact solution of an Ising model on a fractal modular network
• Generating-function approach for bond percolations in hierarchical networks
• Transition-type change between an inverted Berezinskii-Kosterlitz-Thouless transition and an abrupt transition in the bond percolation on a random hierarchical small-world network
• Scaling of Clusters near Discontinuous Percolation Transitions in Hyperbolic Networks
• Generalized scaling theory for critical phenomena including essential singularity and infinite dimensionality
• Criticality governed by the stable renormalization fixed point of the Ising model in the hierarchical small-world network
• From explosive to infinite-order transitions on a hyperbolic network
• Ising model on a hyperbolic plane with a boundary
• Absence of the non-percolating phase for percolation on the non-planar Hanoi network

Cited by in corpus (12)

• Topological Percolation on Hyperbolic Simplicial Complexes
• Percolation on branching simplicial and cell complexes and its relation to interdependent percolation
• Lower-Critical Spin-Glass Dimension from 23 Sequenced Hierarchical Models
• The Chiral Potts Spin Glass in d=2 and 3 Dimensions
• Renormalization group for link percolation on planar hyperbolic manifolds
• Phase Transitions between Different Spin-Glass Phases and between Different Chaoses in Quenched Random Chiral Systems
• Devil's Staircase Continuum in the Chiral Clock Spin Glass with Competing Ferromagnetic-Antiferromagnetic and Left-Right Chiral Interactions
• Renormalization group theory of percolation on pseudo-fractal simplicial and cell complexes
• Stepwise Positional-Orientational Order and the Multicritical-Multistructural Global Phase Diagram of the s=3/2 Ising Model from Renormalization-Group Theory
• Real-Space Renormalization Group for Spectral Properties of Hierarchical Networks
• Evaluation of Spectral Zeta-Functions with the Renormalization Group
• Renormalization-group theory of the abnormal singularities at the critical-order transition in bond percolation on pointed hierarchical graphs