Random Walk And Average Trapping Time On Limited Sierpinski Gasket Network And Its Extended Network

This paper mainly studies the related properties of random walk on the Sierpinski gasket network and its extended model.Random walk is one of the most important dynamic properties on complex networks,and a common feature in random walk problems is the average trapping time.In addition,the self-similar structure of the network is a special property that connects the local topological structure and the global structural characteristics.In recent years,it has become a hot issue in the field of complex networks to further explore the influence of self-similar structures on the property of random walks by studying the average trapping time on regular network models with self-similar structures.The Sierpinski gasket is the most classic selfsimilar network model.This paper uses the Sierpinski network model and its extended network model to explore the dynamic characteristics of random walks on networks with self-similar structures.First,in this paper,based on the self-similar structural characteristics of Sierpinski gaskets,we derive the analytical expression of the average trapping time on it,and explain the reason why this method cannot be extended to high-order Sierpinski gasket networks.Then,using the method of path segmentation,we solved the analytical expression of the average trapping time on the third-order Sierpinski gasket network.The numerical simulation results show that both of them increase exponentially with the number of iterations,and the average trapping time on the third-order network increases faster.In addition,the method can be extended to any high-order Sierpinski gasket network.Then,in order to discuss the effect of the local self-similar structure on the random walk process on the network,we construct a half Sierpinski gasket network model,in which the global self-similar structure of the network model is destroyed and some local self-similar structures are retained.By constructing the auxiliary network,we obtain the analytical expression of the average trapping time on the half Sierpinski gasket network.Numerical simulation results show that the average trapping time of the network model is basically consistent with the original Sierpinski gasket network.This also proves that compared with the strict global self-similar structure,the local selfsimilar structure still maintains the influence on the random walk process.Finally,in order to explore the relationship between the average trapping time on the local self-similar network and the stitching mode or size of the local self-similar structure,we construct the joint Sierpinski gasket network model and the horizontal Sierpinski gasket network model respectively.By stitching variables and matrix method,we obtain the analytical expression of the average trapping time on the joint Sierpinski gasket network model.The numerical simulation results of the three kinds of incomplete Sierpinski networks constructed by this model show that the stitching method does affect the numerical results of the average trapping time under the same number of iterations,but does not change the exponential increasing law between the analytic expression and the number of iterations.Then,by using the structural characteristics of the auxiliary network and the properties of the resistance network,we obtain the analytical expression of the average trapping time in the horizontal Sierpinski gasket network model.The numerical simulation results show that the size of local selfsimilar structures will directly restrict the effect of self-similar structures on the random walk properties of networks.