Multifractal Analysis And Laplace Spectrum Of Horizontal Visibility Networks Constructed From Fractional Brownian Motions

In this work, using the horizontal visibility graph algorithm, we build the hor-izontal visibility networks from fractional Brownian motions (FBMs) with differcnt Hurst index,and then study the properties of these networks. It has been proved that the horizontal visibility network inherits the fractal property in the fractional Brownian motion. We aim to study whether the horizontal visibility network has multifractality property and to analyze the property of Laplace spectrum of these network.First, we used the sandbox algorithm to sduty the multifractality of horizontal visibility network, and tried to find out the relationship between this property and Hurst index H. By sandbox algorithm, we got the fractal dimension of the networks < dB> and found that its approximate expression with H is dB= 2.0121 - 1.0256H, what is close to the theoretical solution 2 - H. We found multifractality exists in horizontal visibility networks constructed from fractional Brownians, and the multifractality of the networks tend to be weakened when the Hurst index H become larger. The average correlation dimension and information dimension of the network decrease with the increase of HAfter that, we studied the second-smallest eigenvalue, the third-smallest eigen-value, and the maximum eigenvalue of the Laplacian for the horizontal visibility networks. We found that the second-smallest eigenvalue, the third-smallest eigen-values of the networks decrease with the increase of H, has a linear relation with the H. Meanwhile, the energy of the networks becomes lower with the increase of H.