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A general model of hierarchical fractal scale-free networks. PLoS One 2022; 17:e0264589. [PMID: 35312679 PMCID: PMC8936503 DOI: 10.1371/journal.pone.0264589] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2021] [Accepted: 02/11/2022] [Indexed: 11/19/2022] Open
Abstract
We propose a general model of unweighted and undirected networks having the scale-free property and fractal nature. Unlike the existing models of fractal scale-free networks (FSFNs), the present model can systematically and widely change the network structure. In this model, an FSFN is iteratively formed by replacing each edge in the previous generation network with a small graph called a generator. The choice of generators enables us to control the scale-free property, fractality, and other structural properties of hierarchical FSFNs. We calculate theoretically various characteristic quantities of networks, such as the exponent of the power-law degree distribution, fractal dimension, average clustering coefficient, global clustering coefficient, and joint probability describing the nearest-neighbor degree correlation. As an example of analyses of phenomena occurring on FSFNs, we also present the critical point and critical exponents of the bond-percolation transition on infinite FSFNs, which is related to the robustness of networks against edge removal. By comparing the percolation critical points of FSFNs whose structural properties are the same as each other except for the clustering nature, we clarify the effect of the clustering on the robustness of FSFNs. As demonstrated by this example, the present model makes it possible to elucidate how a specific structural property influences a phenomenon occurring on FSFNs by varying systematically the structures of FSFNs. Finally, we extend our model for deterministic FSFNs to a model of non-deterministic ones by introducing asymmetric generators and reexamine all characteristic quantities and the percolation problem for such non-deterministic FSFNs.
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Dos Santos CTG, Vieira AP, Salinas SR, Andrade RFS. Real-space renormalization-group treatment of the Maier-Saupe-Zwanzig model for biaxial nematic structures. Phys Rev E 2021; 103:032111. [PMID: 33862734 DOI: 10.1103/physreve.103.032111] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/06/2020] [Accepted: 02/11/2021] [Indexed: 11/07/2022]
Abstract
The Maier-Saupe-Zwanzig model for the nematic phase transitions in liquid crystals is investigated in a diamond hierarchical lattice. The model takes into account a parameter to describe the biaxiality of the microscopic units. Also, a suitably chosen external field is added to the Hamiltonian to allow the determination of critical parameters associated with the nematic phase transitions. Using the transfer-matrix technique, the free energy and its derivatives are obtained in terms of recursion relations between successive generations of the hierarchical lattice. In addition, a real-space renormalization-group approach is developed to obtain the critical parameters of the same model system. Results of both methods are in excellent agreement. There are indications of two continuous phase transitions. One of them corresponds to a uniaxial-isotropic transition, in the class of universality of the three-state Potts model on the diamond hierarchical lattice. The transition between the biaxial and the uniaxial phases is in the universality class of the Ising model on the same lattice.
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Affiliation(s)
- Cícero T G Dos Santos
- Instituto de Física, Universidade Federal da Bahia, 40170-115 Salvador, Brazil.,Instituto Federal de Educação, Ciência e Tecnologia do Sertão Pernambucano, 56302-100 Petrolina, Brazil
| | - André P Vieira
- Universidade de Sao Paulo, Instituto de Fisica, Rua do Matao, 1371, 05508-090 Sao Paulo, SP, Brazil
| | - Silvio R Salinas
- Universidade de Sao Paulo, Instituto de Fisica, Rua do Matao, 1371, 05508-090 Sao Paulo, SP, Brazil
| | - Roberto F S Andrade
- Instituto de Física, Universidade Federal da Bahia, 40170-115 Salvador, Brazil.,Centre for Data and Knowledge Integration for Health (CIDACS), Instituto Gonçalo Moniz, Fundação Oswaldo Cruz, 41745-715 Salvador, Brazil
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Le JX, Yang ZR. Critical slowing down of the kinetic Gaussian model on hierarchical lattices. Phys Rev E 2005; 71:016105. [PMID: 15697656 DOI: 10.1103/physreve.71.016105] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/17/2004] [Revised: 09/30/2004] [Indexed: 11/07/2022]
Abstract
The critical slowing down of the kinetic Gaussian model on hierarchical lattices is studied by means of a real-space time-dependent renormalization-group transformation. The dynamic critical exponent z and the exponent Delta are calculated. For hierarchical lattices with reducible generators, both the dynamic critical exponent z and the exponent Delta are independent of the fractal dimension Df of the lattice, the number of branches m, and the number of bonds per branch b of the generator--i.e., z = 2 and Delta = 1. For hierarchical lattices with irreducible generators, the exponent Delta is the same--i.e., Delta = 1; however, the dynamic critical exponent z is dependent on the concrete geometrical structure of these lattices. In addition, it was found that the lattice dependence of the correlation-length critical exponent nu is the same as that of the dynamic critical exponent z. Finally we give a brief discussion about universality for critical dynamics.
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Affiliation(s)
- Jian-Xin Le
- CCAST World Laboratory, Box 8730, Beijing 100080, China.
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Le JX, Yang ZR. Phase transitions of the Ashkin-Teller model including antiferromagnetic interactions on a type of diamond hierarchical lattice. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2004; 69:066107. [PMID: 15244667 DOI: 10.1103/physreve.69.066107] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/27/2003] [Indexed: 05/24/2023]
Abstract
Using the real-space renormalization-group transformation, we study the phase transitions of the Ashkin-Teller model including the antiferromagnetic interactions on a type of diamond hierarchical lattices, of which the number of bonds per branch of the generator is odd. The isotropic Ashkin-Teller model and the anisotropic one are, respectively, investigated. We find that the phase diagram, for the isotropic Ashkin-Teller model, consists of five phases, two of which are associated with the partially antiferromagnetic ordering of the system, while the phase diagram, for the anisotropic Ashkin-Teller model, contains 11 phases, six of which are related to the partially antiferromagnetic ordering of the system. The correlation length critical exponents and the crossover exponents are also calculated.
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Affiliation(s)
- Jian-Xin Le
- Department of Physics and Institute of Theoretical Physics, Beijing Normal University, Beijing 100875, China.
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Le JX, Yang ZR. Phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2003; 68:066105. [PMID: 14754267 DOI: 10.1103/physreve.68.066105] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2003] [Indexed: 05/24/2023]
Abstract
The phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices is studied by means of the transfer-matrix method and the real-space renormalization-group transformation. We find that the phase diagram, for the ferromagnetic case, consists of five phases, i.e., the fully disordered paramagnetic phase P, the fully ordered ferromagnetic phase F, and three partially ordered ferromagnetic phases F(s), F(sigma), and F(s sigma), as well as ten nontrivial fixed points. The correlation length critical exponents and the crossover exponents are also calculated. In addition, we also investigate the variations of the critical exponents with the fractal dimension d(f), the number of branches m, and the number of bonds per branch b of the generator of the family of diamond-type hierarchical lattices. Finally we give a brief discussion about universality.
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Affiliation(s)
- Jian-Xin Le
- Department of Physics and Institute of Theoretical Physics, Beijing Normal University, Beijing 100875, China.
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Redinz JA. Ising ferromagnet on a fractal family: Thermodynamical functions and scaling laws. PHYSICAL REVIEW. B, CONDENSED MATTER 1995; 51:2930-2936. [PMID: 9979069 DOI: 10.1103/physrevb.51.2930] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Wu Y, Qin Y, Yang ZR. Diluted Potts model on diamond-type hierarchical lattices. PHYSICAL REVIEW. B, CONDENSED MATTER 1993; 48:3171-3176. [PMID: 10008740 DOI: 10.1103/physrevb.48.3171] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Potts model on infinitely ramified Sierpinski-gasket-type fractals and algebraic order at antiferromagnetic phases. PHYSICAL REVIEW. B, CONDENSED MATTER 1992; 46:11642-11656. [PMID: 10003053 DOI: 10.1103/physrevb.46.11642] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/12/2023]
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Yung Q, Yang ZR. Crossover from a fractal lattice to a Euclidean lattice for the thermodynamic properties of a triplet-interaction Ising model. PHYSICAL REVIEW. B, CONDENSED MATTER 1991; 43:13342-13347. [PMID: 9997164 DOI: 10.1103/physrevb.43.13342] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Inayat-Hussain AA, Buckingham MJ. Continuously varying critical exponents to O(1/n). PHYSICAL REVIEW. A, ATOMIC, MOLECULAR, AND OPTICAL PHYSICS 1990; 41:5394-5417. [PMID: 9902926 DOI: 10.1103/physreva.41.5394] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
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Kaufman M, Kahana M. Cayley-tree Ising model with antiferromagnetic nearest-neighbor and ferromagnetic equivalent-neighbor interactions. PHYSICAL REVIEW. B, CONDENSED MATTER 1988; 37:7638-7642. [PMID: 9944060 DOI: 10.1103/physrevb.37.7638] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Southern BW, Kneevic M. Zeros of the partition function of Ising models on fractal lattices. PHYSICAL REVIEW. B, CONDENSED MATTER 1987; 35:5036-5042. [PMID: 9940686 DOI: 10.1103/physrevb.35.5036] [Citation(s) in RCA: 12] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Wu Y, Hu B. Phase transitions on complex Sierpin-acuteski carpets. PHYSICAL REVIEW. A, GENERAL PHYSICS 1987; 35:1404-1411. [PMID: 9898282 DOI: 10.1103/physreva.35.1404] [Citation(s) in RCA: 32] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Hauser PR, Saxena VK. Geometrical phase transitions on hierarchical lattices and universality. PHYSICAL REVIEW. B, CONDENSED MATTER 1986; 34:8193-8195. [PMID: 9939526 DOI: 10.1103/physrevb.34.8193] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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Melrose JR. Comment on "Problem of universality in phase transitions on hierarchical lattices". PHYSICAL REVIEW LETTERS 1986; 57:2601. [PMID: 10033811 DOI: 10.1103/physrevlett.57.2601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/23/2023]
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Lee JC. Randomly closed Cayley trees and fractal dimensionality. PHYSICAL REVIEW. B, CONDENSED MATTER 1986; 34:3297-3302. [PMID: 9940068 DOI: 10.1103/physrevb.34.3297] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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Hu B. Extended universality and the question of spin on hierarchical lattices. PHYSICAL REVIEW. B, CONDENSED MATTER 1986; 33:6503-6504. [PMID: 9939211 DOI: 10.1103/physrevb.33.6503] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 04/11/2023]
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