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

Abstract

Abstract.In this paper, we study random walks on a small-world scale-free network, also called as pseudofractal scale-free web (PSFW), and analyze the volatilities of first passage time (FPT) and first return time (FRT) by using the variance and the reduced moment as the measures. Note that the FRT and FPT are deeply affected by the starting or target site. We don't intend to enumerate all the possible cases and analyze them. We only study the volatilities of FRT for a given hub (i.e., node with highest degree) and the volatilities of the global FPT (GFPT) to a given hub, which is the average of the FPTs for arriving at a given hub from any possible starting site selected randomly according to the equilibrium distribution of the Markov chain. Firstly, we calculate exactly the probability generating function of the GFPT and FRT based on the self-similar structure of the PSFW. Then, we calculate the probability distribution, the mean, the variance and reduced moment of the GFPT and FRT by using the generating functions as a tool. Results show that: the reduced moment of FRT grows with the increasing of the network order N and tends to infinity while N → ∞; but for the reduced moments of GFPT, it is almost a constant(≈ 1.1605) for large N . Therefore, on the PSFW of large size, the FRT has huge fluctuations and the estimate provided by MFRT is unreliable, whereas the fluctuations of the GFPT is much smaller and the estimate provided by its mean is more reliable. The method we propose can also be used to analyze the volatilities of FPT and FRT on other networks with selfsimilar structure, such as (u, v) flowers and recursive scale-free trees.Volatilities of FPT and FRT on a small-world scale-free network 2 IntroductionFirst passage time (FPT), which is the time it takes a random walker to reach a given site for the first time, and first return time (FRT), which is the time it takes a random walker to return to the starting site for the first time, are two important quantities in the random walk literature [1][2][3][4]. The importance lies in the fact that many physical processes are controlled by first passage events [5][6][7][8][9][10][11][12], and that FRT can model the time intervals between two successive extreme events [13][14][15][16][17], such as traffic jams in roads, the floods, the droughts, and power blackouts in electrical power grid, etc [18][19][20]. Both FPT and FRT are random variables which can not be determined exactly and researchers can only try to find suitable quantities to estimate them. A first step consists in the analysis of the mean of the two random variables, the mean first-passage time (MFPT) and the mean first return time (MFRT). For the MFRT, it can be calculated from the stationary distribution directly. That is to say, the MFRT of node v is 2m/d v on any finite connected network, where m is the total numbers of edges of the network and d v is the degree of node v [21,22]. For the MFPT, general formula to calculate the MPFT between any two nodes in any finite networks was presented [...

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