Multifractal Measures on Small-World Networks

Abstract

We investigate the multifractals of the normalized first passage time on one-dimensional small-world network with both reflecting and absorbing barriers. The multifractals is estimated from the distribution of the normalized first passage time characterized by the random walk on the small-world network with three fractions of edges rewired randomly. Particularly, our estimate is the fractal dimension D 0 = 0. 917, 0.926, 0.930 for lattice points L = 80 and a randomly rewired fraction p = 0.2. The numerical result is found to disappear multifractal properties in the regime On the other hand, one of stochastic processes for regular and disordered systems includes the random walk theory [10], and a line of development of importance for the advancement of the random walk is the continuous-time random walk theory, formerly introduced by Montroll and Weiss [11], which substantially elaborated on the transition probability and the distribution of the pausing times [12]. To describe quantitatively hopping of the continuous-time random walk problem, one has to know how to calculate both the transition probability and random pausing time of the random walker between lattice points. In particular, the transition rate in charge conductions of the disordered solid has depended on the spatial separation between the localized centers and on the energy difference between the initial and final state [13]. Until now, the continuous-time random walk theory mainly has studied in natural and social sciences, and among the outstanding topics, reaction and strange kinetics [14], fractional diffusion equations [15], random networks, earthquake modeling, hydrology, and financial options [16] have been the focus of attention to many researchers. 2The mean first passage time for the random walk theory is the statistical quantity that defines the average time arriving at the absorbing barrier for the first time.Particularly, Sinai model [17,18] has discussed the first passage time related to the random barrier with the absorbing and reflecting barriers. The Sinai model and other previous works [19] were studied for the mean and mean square displacements dependent anomalously on time. Many stochastic processes on one-dimensional lattice have argued with the mean first passage time, for the specific cases of the transition probability such as chaotic orbits of the logistic and Kim-Kong maps [20]. Recently, Kim and Kong [21] have used the box-counting method to estimate the generalized dimension and the scaling exponent for the mountain height and the sea-bottom depth.In this paper, we consider a randomly rewired fraction p of one-dimensional small-world networks, so that choosingallows to interpolate between the regular (p = 0) and random (p = 1) networks. The purpose of this paper is to apply the small-world graph theory to the random walk on one-dimensional lattice with both reflecting and absorbing barriers. The multifractals from the normalized first passage time is estimated on small-world networks with three fractions of ed...

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