Volatilities analysis of first-passage time and first-return time on a small-world scale-free network

摘要

In this paper, we study random walks on a small-world scale-free network,also called as pseudofractal scale-free web (PSFW), and analyze thevolatilities of first passage time (FPT) and first return time (FRT) by usingthe variance and the reduced moment as the measures. Note that the FRT and FPTare deeply affected by the starting or target site. We don't intend toenumerate all the possible cases and analyze them. We only study thevolatilities of FRT for a given hub (i.e., node with highest degree) and thevolatilities of the global FPT (GFPT) to a given hub, which is the average ofthe FPTs for arriving at a given hub from any possible starting site selectedrandomly according to the equilibrium distribution of the Markov chain.Firstly, we calculate exactly the probability generating function of the GFPTand FRT based on the self-similar structure of the PSFW. Then, we calculate theprobability distribution, the mean, the variance and reduced moment of the GFPTand FRT by using the generating functions as a tool. Results show that: thereduced moment of FRT grows with the increasing of the network order $N$ andtends to infinity while $N\rightarrow\infty$; but for the reduced moments ofGFPT, it is almost a constant($\approx1.1605$) for large $N$. Therefore, on thePSFW of large size, the FRT has huge fluctuations and the estimate provided byMFRT is unreliable, whereas the fluctuations of the GFPT is much smaller andthe estimate provided by its mean is more reliable. The method we propose canalso be used to analyze the volatilities of FPT and FRT on other networks withself-similar structure, such as $(u, v)$ flowers and recursive scale-freetrees.