{"title":"Universality of critical dynamics on a complex network","authors":"Mrinal Sarkar, Tilman Enss, Nicolò Defenu","doi":"arxiv-2401.00092","DOIUrl":null,"url":null,"abstract":"We investigate the role of the spectral dimension $d_s$ in determining the\nuniversality of phase transitions on a complex network. Due to its structural\nheterogeneity, a complex network generally acts as a disordered system.\nSpecifically, we study the synchronization and entrainment transitions in the\nnonequilibrium dynamics of the Kuramoto model and the phase transition of the\nequilibrium dynamics of the classical $XY$ model, thereby covering a broad\nspectrum from nonlinear dynamics to statistical and condensed matter physics.\nUsing linear theory, we obtain a general relationship between the dynamics\noccurring on the network and the underlying network properties. This yields the\nlower critical spectral dimension of the phase synchronization and entrainment\ntransitions in the Kuramoto model as $d_s=4$ and $d_s=2$ respectively, whereas\nfor the phase transition in the $XY$ model it is $d_s=2$. To test our\ntheoretical hypotheses, we employ a network where any two nodes on the network\nare connected with a probability proportional to a power law of the distance\nbetween the nodes; this realizes any desired $d_s\\in [1, \\infty)$. Our detailed\nnumerical study agrees well with the prediction of linear theory for the phase\nsynchronization transition in the Kuramoto model. However, it shows a clear\nentrainment transition in the Kuramoto model and phase transition in the $XY$\nmodel at $d_s \\gtrsim 3$, not $d_s=2$ as predicted by linear theory. Our study\nindicates that network disorder in the region $2 \\leq d_s \\lesssim 3$ seems to\nbe relevant and have a profound effect on the dynamics.","PeriodicalId":501305,"journal":{"name":"arXiv - PHYS - Adaptation and Self-Organizing Systems","volume":"103 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2023-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - PHYS - Adaptation and Self-Organizing Systems","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2401.00092","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We investigate the role of the spectral dimension $d_s$ 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 $XY$ 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 $d_s=4$ and $d_s=2$ respectively, whereas
for the phase transition in the $XY$ model it is $d_s=2$. 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 $d_s\in [1, \infty)$. 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 $XY$
model at $d_s \gtrsim 3$, not $d_s=2$ as predicted by linear theory. Our study
indicates that network disorder in the region $2 \leq d_s \lesssim 3$ seems to
be relevant and have a profound effect on the dynamics.