We study the decay process for the reaction-diffusion process of three species on the small-world network. The decay process is manipulated from the deterministic rate equation of three species in the reaction-diffusion system. The particle density a
nd the global reaction rate on a two dimensional small-world network adding new random links is discussed numerically, and the global reaction rate before and after the crossover is also found by means of the Monte Carlo simulation. The time-dependent global reaction rate scales as a power law with the scaling exponent 0.66 at early time regime while it scales with -0.50 at long time regime, in all four cases of the added probability $p=0.2-0.8$. Especially, our result presented is compared with the numerical calculation of regular networks.
We study the rank distribution, the cumulative probability, and the probability density of returns of stock prices of listed firms traded in four stock markets. We find that the rank distribution and the cumulative probability of stock prices traded
in are consistent approximately with the Zipfs law or a power law. It is also obtained that the probability density of normalized returns for listed stocks almost has the form of the exponential function. Our results are compared with those of other numerical calculations.
This paper investigates the rank distribution, cumulative probability, and probability density of price returns for the stocks traded in the KSE and the KOSDAQ market. This research demonstrates that the rank distribution is consistent approximately
with the Zipfs law with exponent $alpha = -1.00$ (KSE) and -1.31 (KOSDAQ), similar that of stock prices traded on the TSE. In addition, the cumulative probability distribution follows a power law with scaling exponent $beta = -1.23$ (KSE) and -1.45 (KOSDAQ). In particular, the evidence displays that the probability density of normalized price returns for two kinds of assets almost has the form of an exponential function, similar to the result in the TSE and the NYSE.
We investigate the herd behavior of returns for the yen-dollar exchange rate in the Japanese financial market. It is obtained that the probability distribution $P(R)$ of returns $R$ satisfies the power-law behavior $P(R) simeq R^{-beta}$ with the exp
onents $ beta=3.11$(the time interval $tau=$ one minute) and 3.36($tau=$ one day). The informational cascade regime appears in the herding parameter $Hge 2.33$ at $tau=$ one minute, while it occurs no herding at $tau=$ one day. Especially, we find that the distribution of normalized returns shows a crossover to a Gaussian distribution at one time step $Delta t=1$ day.
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 charac
trized 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 p> p_c, where p_c is the critical rewired fraction.
We propose the deterministic rate equation of three-species in the reaction - diffusion system. For this case, our purpose is to carry out the decay process in our three-species reaction-diffusion model of the form $A+B+Cto D$. The particle density a
nd the global reaction rate are also shown analytically and numerically on a two-dimensional square lattice with the periodic boundary conditions. Especially, the crossover of the global reaction rate is discussed in both early-time and long-time regimes.
The multifractal behavior of the normalized first passage time is investigated on the two dimensional Sierpinski gasket with both absorbing and reflecting barriers. The normalized first passage time for Sinai model and the logistic model to arrive at
the absorbing barrier after starting from an arbitrary site, especially obtained by the calculation via the Monte Carlo simulation, is discussed numerically. The generalized dimension and the spectrum are also estimated from the distribution of the normalized first passage time, and compared with the results on the finitely square lattice.
