Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions

Zu-Guo Yu School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, People’s Republic of China; School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane Q4001, Australia Huan Zhang School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, People’s Republic of China Da-Wen Huang School of Mathematics and Computational Science, Xiangtan University, Hunan 411105, People’s Republic of China Yong Lin Department of Mathematics, School of Information, Remin University of China, Beijing 100872, People’s Republic of China Vo Anh School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane Q4001, Australia

arXiv subject: Statistical Mechanics (cond-mat.stat-mech) mathscidoc:2207.48001

Journal of Statistical Mechanics: Theory and Experiment, 2016, 2016.3
Many studies have shown that additional information can be gained on time series by investigating their associated complex networks. In this work, we investigate the multifractal property and Laplace spectrum of the horizontal visibility graphs (HVGs) constructed from fractional Brownian motions. We aim to identify via simulation and curve fitting the form of these properties in terms of the Hurst index H. First, we use the sandbox algorithm to study the multifractality of these HVGs. It is found that multifractality exists in these HVGs. We find that the average fractal dimension $\langle D(0)\rangle $ of HVGs approximately satisfies the prominent linear formula $\langle D(0)\rangle =2-H$ ; while the average information dimension $\langle D(1)\rangle $ and average correlation dimension $\langle D(2)\rangle $ are all approximately bi-linear functions of H when $H\geqslant 0.15$ . Then, we calculate the spectrum and energy for the general Laplacian operator and normalized Laplacian operator of these HVGs. We find that, for the general Laplacian operator, the average logarithm of second-smallest eigenvalue $\langle \ln \left({{u}_{2}}\right)\rangle $ , the average logarithm of third-smallest eigenvalue $\langle \ln \left({{u}_{3}}\right)\rangle $ , and the average logarithm of maximum eigenvalue $\langle \ln \left({{u}_{n}}\right)\rangle $ of these HVGs are approximately linear functions of H; while the average Laplacian energy $\langle {{E}_{\text{nL}}}\rangle $ is approximately a quadratic polynomial function of H. For the normalized Laplacian operator, $\langle \ln \left({{u}_{2}}\right)\rangle $ and $\langle \ln \left({{u}_{3}}\right)\rangle $ of these HVGs approximately satisfy linear functions of H; while $\langle \ln \left({{u}_{n}}\right)\rangle $ and $\langle {{E}_{\text{nL}}}\rangle $ are approximately a 4th and cubic polynomial function of H respectively.
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@inproceedings{zu-guo2016multifractality,
  title={Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions},
  author={Zu-Guo Yu, Huan Zhang, Da-Wen Huang, Yong Lin, and Vo Anh},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707120431926471554},
  booktitle={Journal of Statistical Mechanics: Theory and Experiment},
  volume={2016},
  year={2016},
}
Zu-Guo Yu, Huan Zhang, Da-Wen Huang, Yong Lin, and Vo Anh. Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions. 2016. Vol. 2016. In Journal of Statistical Mechanics: Theory and Experiment. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220707120431926471554.
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