The first return time (FRT) is the time it takes a random walker to first return to its original site, and the global first passage time (GFPT) is the first passage time for a random walker to move from a randomly selected site to a given site. We find that in finite networks, the variance of FRT, Var(FRT), can be expressed as Var(FRT) = 2⟨FRT⟩⟨GFPT⟩ - ⟨FRT⟩ - ⟨FRT⟩, where ⟨·⟩ is the mean of the random variable. Therefore a method of calculating the variance of FRT on general finite networks is presented. We then calculate Var(FRT) and analyze the fluctuation of FRT on regular branched networks (i.e., Cayley tree) by using Var(FRT) and its variant as the metric. We find that the results differ from those in such other networks as Sierpinski gaskets, Vicsek fractals, T-graphs, pseudofractal scale-free webs, (u, v) flowers, and fractal and non-fractal scale-free trees.
The global first passage time (GFPT) is the first-passage time for a random walker from a randomly selected site to a given site. Here, we find the exact relation between the moments of GFPT and those of first return time (FRT) on general finite networks firstly. The exact relation is meaningful for understanding the dynamic taken place on the networks. It is also helpful to simplify the simulation of random walk on the networks. Then we derive the exact results for the first and second moments, together with asymptotic results for the higher moments, of the GFPT and FRT to a boundary node on the treelike fractal. We find that nth () moments of the GFPT and the FRT scale with the network size N as: and , where , denote the nth moments of the GFPT and the FRT respectively, ds is the spectral dimension of the network.
Mineralized intensity is one of the key problems in the study of ore deposit. The mineralization degree in drifts can be categorized into non-mineralized, weakly mineralized, mineralized ranks based on the number of the gold grades greater than cut-off. In this paper, the lacunarity analysis is utilized to analyze the mineralized intensity in drifts of the Dayingezhuang gold ore deposit in Jiaodong, Shandong province, China. It shows that with the increase of mineralized rank, the lacunarity index becomes greater. The results will provide information for both the ore-forming process and the exploitation.
Random walks have wide application in real lives, such as target search, reaction kinetics, polymer chains, and so on. In this paper, we consider discrete random walks on general connected networks and focus on the global mean first return time (GMFRT), which is defined as the mean first return time averaged over all the possible starting positions (vertices), aiming at finding the structures which have the maximal (or the minimal) GMFRT. Our results show that, among all trees with a given number of vertices, trees with linear structure are those with the minimal GMFRT and stars are those with the maximal GMFRT. We also find that, among all unweighted and undirected connected simple graphs with a given number of edges and vertices, the graphs maximizing (resp. minimizing) the GMFRT are the ones for which the variance of the nodes degrees is the largest (resp. the smallest).
Random walks have wide application in real lives, ranging from target search, reaction kinetics, polymer chains, to the forecast of the arrive time of extreme events, diseases or opinions. In this paper, we consider discrete random walks on general connected networks and focus on the analysis of the global mean first return time (GMFRT), which is defined as the mean first return time averaged over all the possible starting positions (vertices), aiming at finding the structures who have the maximal (or the minimal) GMFRT among all connected graphs with the same number of vertices and edges. Our results show that, among all trees with the same number of vertices, trees with linear structure are the structures with the minimal GMFRT and stars are the structures with the maximal GMFRT. We also find that, among all connected graphs with the same number of vertices, the graphs whose vertices have the same degree, are the structures with the minimal GMFRT; and the graphs whose vertex degrees have the biggest difference, are the structures with the maximal GMFRT. We also present the methods for constructing the graphs with the maximal GMFRT (or the minimal GMFRT), among all connected graphs with the same number of vertices and edges.
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