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

Abstract

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) 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(u 2 ) , the average logarithm of third-smallest eigenvalue ln(u 3 ) , and the average logarithm of maximum eigenvalue ln(u n ) of these HVGs are approximately linear functions of H; while the average Laplacian energy E nL is approximately a quadratic polynomial function of H. For the normalized Laplacian operator, ln(u 2 ) and ln(u 3 ) of these HVGs approximately satisfy linear functions of H; while ln(u n ) and E nL are approximately a 4th and cubic polynomial function of H respectively.

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