Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions
Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions
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DOI：
10.1088/1742-5468/2016/03/033206
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发表时间：
2016-02
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影响因子：
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通讯作者：
Zuguo Yu;Huan Zhang;Da-Wen Huang;Yong Lin;V. Anh
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文献类型：
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作者：
Zuguo Yu;Huan Zhang;Da-Wen Huang;Yong Lin;V. Anh
许多研究表明，通过调查时间序列相关的复杂网络，可以获得额外的信息。本文研究了由分数阶布朗运动构造的水平可见性图的多重分形性质和拉普拉斯谱。我们的目标是通过模拟和曲线拟合来确定这些属性在Hurst指数h方面的形式。首先，我们使用沙盒算法来研究这些hvg的多重分形。发现在这些hvg中存在多重分形。我们发现平均分形维数⟨D(0)⟩？hvg的>近似满足⟨D(0)⟩=2 - H ?>的显著线性公式；而平均信息维⟨D(1)⟩？>和平均相关维⟨D(2)⟩？当H小于0.15 ?>时，>都是H的近似双线性函数。然后，我们计算了这些hvg的一般拉普拉斯算子和归一化拉普拉斯算子的谱和能量。我们发现，对于一般的拉普拉斯算子，第二小特征值⟨ln （u2）⟩的平均对数？>，第三小特征值⟨ln （u3）⟩的平均对数？>，和最大特征值⟨ln （un）⟩的平均对数？这些hvg的>是H的近似线性函数；而拉普拉斯的平均能量⟨EnL⟩？>近似为h的二次多项式函数。对于归一化拉普拉斯算子，⟨ln （u2）⟩？>和⟨ln （u3）⟩？这些hvg的>近似满足H的线性函数；而⟨ln （un）⟩？>和⟨EnL⟩？>分别近似为H的四次和三次多项式函数。
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 ⟨D(0)⟩ ?> of HVGs approximately satisfies the prominent linear formula ⟨D(0)⟩=2−H ?>; while the average information dimension ⟨D(1)⟩ ?> and average correlation dimension ⟨D(2)⟩ ?> are all approximately bi-linear functions of H when H⩾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 ⟨ln(u2)⟩ ?>, the average logarithm of third-smallest eigenvalue ⟨ln(u3)⟩ ?>, and the average logarithm of maximum eigenvalue ⟨ln(un)⟩ ?> of these HVGs are approximately linear functions of H; while the average Laplacian energy ⟨EnL⟩ ?> is approximately a quadratic polynomial function of H. For the normalized Laplacian operator, ⟨ln(u2)⟩ ?> and ⟨ln(u3)⟩ ?> of these HVGs approximately satisfy linear functions of H; while ⟨ln(un)⟩ ?> and ⟨EnL⟩ ?> are approximately a 4th and cubic polynomial function of H respectively.