{"title":"Universality of critical dynamics on a complex network","authors":"Mrinal Sarkar, Tilman Enss, Nicolò Defenu","doi":"10.1103/physrevb.110.014208","DOIUrl":null,"url":null,"abstract":"We investigate the role of the spectral dimension <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><msub><mi>d</mi><mi>s</mi></msub></math> in determining the universality of phase transitions on a complex network. Due to its structural heterogeneity, a complex network generally acts as a disordered system. Specifically, we study the synchronization and entrainment transitions in the nonequilibrium dynamics of the Kuramoto model and the phase transition of the equilibrium dynamics of the classical <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>X</mi><mi>Y</mi></mrow></math> model, thereby covering a broad spectrum from nonlinear dynamics to statistical and condensed matter physics. Using linear theory, we obtain a general relationship between the dynamics occurring on the network and the underlying network properties. This yields the lower critical spectral dimension of the phase synchronization and entrainment transitions in the Kuramoto model as <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>d</mi><mi>s</mi></msub><mo>=</mo><mn>4</mn></mrow></math> and <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>d</mi><mi>s</mi></msub><mo>=</mo><mn>2</mn></mrow></math>, respectively, whereas for the phase transition in the <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>X</mi><mi>Y</mi></mrow></math> model it is <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>d</mi><mi>s</mi></msub><mo>=</mo><mn>2</mn></mrow></math>. To test our theoretical hypotheses, we employ a network where any two nodes on the network are connected with a probability proportional to a power law of the distance between the nodes; this realizes any desired <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>d</mi><mi>s</mi></msub><mo>∈</mo><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow></mrow></math>. Our detailed numerical study agrees well with the prediction of linear theory for the phase synchronization transition in the Kuramoto model. However, it shows a clear entrainment transition in the Kuramoto model and phase transition in the <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mi>X</mi><mi>Y</mi></mrow></math> model at <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>d</mi><mi>s</mi></msub><mo>≳</mo><mn>3</mn></mrow></math>, not <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><msub><mi>d</mi><mi>s</mi></msub><mo>=</mo><mn>2</mn></mrow></math> as predicted by linear theory. Our study indicates that network disorder in the region <math xmlns=\"http://www.w3.org/1998/Math/MathML\"><mrow><mn>2</mn><mo>≤</mo><msub><mi>d</mi><mi>s</mi></msub><mo>≲</mo><mn>3</mn></mrow></math> introduces strong finite-size fluctuations, which makes it extremely difficult to probe the existence of the ordered phase as predicted, affecting the dynamics profoundly.","PeriodicalId":20082,"journal":{"name":"Physical Review B","volume":null,"pages":null},"PeriodicalIF":3.7000,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Physical Review B","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1103/physrevb.110.014208","RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"Physics and Astronomy","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate the role of the spectral dimension in determining the universality of phase transitions on a complex network. Due to its structural heterogeneity, a complex network generally acts as a disordered system. Specifically, we study the synchronization and entrainment transitions in the nonequilibrium dynamics of the Kuramoto model and the phase transition of the equilibrium dynamics of the classical model, thereby covering a broad spectrum from nonlinear dynamics to statistical and condensed matter physics. Using linear theory, we obtain a general relationship between the dynamics occurring on the network and the underlying network properties. This yields the lower critical spectral dimension of the phase synchronization and entrainment transitions in the Kuramoto model as and , respectively, whereas for the phase transition in the model it is . To test our theoretical hypotheses, we employ a network where any two nodes on the network are connected with a probability proportional to a power law of the distance between the nodes; this realizes any desired . Our detailed numerical study agrees well with the prediction of linear theory for the phase synchronization transition in the Kuramoto model. However, it shows a clear entrainment transition in the Kuramoto model and phase transition in the model at , not as predicted by linear theory. Our study indicates that network disorder in the region introduces strong finite-size fluctuations, which makes it extremely difficult to probe the existence of the ordered phase as predicted, affecting the dynamics profoundly.
期刊介绍:
Physical Review B (PRB) is the world’s largest dedicated physics journal, publishing approximately 100 new, high-quality papers each week. The most highly cited journal in condensed matter physics, PRB provides outstanding depth and breadth of coverage, combined with unrivaled context and background for ongoing research by scientists worldwide.
PRB covers the full range of condensed matter, materials physics, and related subfields, including:
-Structure and phase transitions
-Ferroelectrics and multiferroics
-Disordered systems and alloys
-Magnetism
-Superconductivity
-Electronic structure, photonics, and metamaterials
-Semiconductors and mesoscopic systems
-Surfaces, nanoscience, and two-dimensional materials
-Topological states of matter