由分数布朗运动构造的水平可见图的多重分形性和Laplace谱
Multifractality and Laplace spectrum of horizontal visibility graphs constructed from fractional Brownian motions
摘要：
许多研究表明，通过研究与时间序列相关的复杂网络，可以获得更多的信息。本文研究了由分数布朗运动构造的水平可见图的多重分形性质和Laplace谱。我们的目标是通过模拟和曲线拟合的形式确定这些属性的赫斯特指数$H$。首先，我们使用沙箱算法来研究这些HVG的多重分形。结果表明，这些HVG都存在多重分形。我们发现HVG的平均分形维数$\langle D(0)\rangle$近似满足显式线性公式$\langle D(0)\rangle = 2 - H$ ;而当$H\ge 0.15$时，平均信息维数$\langle D(1)\rangle$和平均关联维数$\langle D(2)\rangle$都近似是$H$的双线性函数.然后，我们计算了这些HVG的广义Laplacian算子和归一化Laplacian算子的谱和能量。我们发现，对于一般拉普拉斯算子，这些HVG的次小特征值$\langle \ln (u_2) \rangle$的平均对数、第三小特征值$\langle \ln (u_3) \rangle$的平均对数和最大特征值$\langle \ln (u_n) \rangle$的平均对数近似为$H$的线性函数，而平均拉普拉斯能量$\langle E_{nL} \rangle$近似为$H$的二次多项式函数.对于归一化拉普拉斯算子，这些HVG的$\langle \ln (u_2) \rangle$和$\langle \ln (u_3) \rangle$近似满足$H$的线性函数，而$\langle \ln (u_n) \rangle$和$\langle E_{nL} \rangle$分别近似为$H$的四次和三次多项式函数.
英文摘要：
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 $H\ge 0.15$. 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 $\langle \ln (u_2) \rangle$, the average logarithm of third-smallest eigenvalue $\langle \ln (u_3) \rangle$, and the average logarithm of maximum eigenvalue $\langle \ln (u_n) \rangle$ of these HVGs are approximately linear functions of $H$; while the average Laplacian energy $\langle E_{nL} \rangle$ is approximately a quadratic polynomial function of $H$. For the normalized Laplacian operator, $\langle \ln (u_2) \rangle$ and $\langle \ln (u_3) \rangle$ of these HVGs approximately satisfy linear functions of $H$; while $\langle \ln (u_n) \rangle$ and $\langle E_{nL} \rangle$ are approximately a 4th and cubic polynomial function of $H$ respectively.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
