Hitting time for random walks on the Sierpinski network and the half Sierpinski network

Abstract

We consider the unbiased random walk on the Sierpinski network (Sn◦N) and the half Sierpinski network (HSn◦N), where n is the generation. Different from the existing works on the Sierpinski gasket, Sn◦N is generated by the nested method and HSn◦N is half of Sn◦N based on the vertical cutting of the symmetry axis. We study the hitting time on Sn◦N and HSn◦N. According to the complete symmetry and structural properties of Sn◦N, we derive the exact expressions of the hitting time on the nth generation of Sn◦N and HSn◦N. The curves of the hitting time for the two networks are almost consistent when n is large enough. The result indicates that the diffusion efficiency of HSn◦N has not changed greatly compared with Sn◦N at a large scale.

Average trapping time on horizontally divided 3-dimensional 3-level Sierpinski gasket network

“…Qi Yi et al [17] obtained several expressions for the hit time of random wandering on Sierpinski and hierarchical graph. Previous studies have been based on two-dimensional cut network or three-dimensional complete network model [18,19]. In addition, considering the effect of local self-similar structures on ATT in three dimensions, researchers [20] investigated the trap problem of three-dimensional cut network.…”

The average trapping time of non-nearest-neighbor jumps on nested networks

“…The diffusion efficiency is inversely proportional to the value of ATT, that is, the smaller the ATT, the higher the diffusion efficiency of the network; the larger the ATT, the lower the diffusion efficiency of the network. Relevant scholars have studied the trapping problem in triangle network [17], crystal network [22], flower network [23,24], ring tree network [25,26], fractal network [27], SG network [28], etc., and obtained the analytical formula of ATT. The walking style of these problems considers a simple random walk, that is, moving from a node to its nearest neighbor (NN) node.…”

“…Based on the 2-level Sierpinski Gasket proposed by Sun et al [28], we consider the construction of a 3-level network and the hybrid walk. In order to find the ATT of the network under this walk, we introduce FPT, FRT and GFPT, which respectively represent the first passage trapping time (FPT), which is the time taken by the random walker to reach the given trap node for the first time, and the first return time (FRT), which is the time taken by the random walker to return to the starting node for the first time, and the global first passage time (GFPT), which is the time of first passage from a randomly selected node to a given node.…”