Research On Random Walks With Trap On Two Kinds Of Self-similar Complex Networks

Trapping problem is closely related to diverse dynamic processes occurring on the network and is an important problem in the field of random walks.Designing appropriate techniques to effectively control the trapping processes and improve the trapping efficiency is an important sub-problem in the study of trapping problem on complex networks.In this paper,we mainly study random walks with trap on two kinds of self-similar complex networks,and focus on the special case where trap positioned at an fixed node.Considering the average trapping time is a common indicator to measure the trapping efficiency,for the trapping problem studied,we derive both analytically and dominating scaling the average trapping time(ATT).The specific work is summarized as follows:In the first work,we mainly study delayed random walks with a trap positioned at an initial node of the Apollonian network.The method dominate the trapping process by varying the transition probability of random walks,and the transition probability is modified by an introduced stochastic parameter.We further obtain the average trapping time(ATT)as a measure of trapping efficiency,and the obtained analytical expressions are in good agreement with the corresponding numerical solutions.The results show that ATT increases sublinearly with the network size when 0?p?1,and the stochastic parameter p only modifies the prefactor and keeps the leading scaling unchangedThe second work mainly studies two types of random walks with a trap on weighted directed networks,namely standard random walks and mixed random walks.The weighted directed network studied can be extended from the previous undirected network as follows:each edge of the original undirected network is regarded as two directed edges with different edge weights,and the weight of each directed edge is controlled by the weight parameter w Finally,the analytical results of the average trapping time for the two types of random walks on the network are obtained respectively,and the results are in good agreement with the nu-merical results for the different values of w and ?.Among them,the first part mainly studies the standard random walk on the weighted directed treelike network.For the standard ran-dom walks on the weighted directed network,the dominant scaling of the average trapping time is completely controlled by the weight parameter w.By adjusting the parameter,the dominant scaling of the average trapping time can be a superlinear function of the network scale.The second part focuses on the mixed random walk on a weighted directed treelike network.The mixed random walk on the weighted directed treelike network is mainly con-trolled by the weight parameter w and the probability parameter ?.The results show that the weight parameter w determines the dominating scaling of ATT,and that parameters w and ?together determine the prefactor.Therefore,in order to obtain the desired leading scalings and prefactor simultaneously,one can first adjust the parameter w to get the desired leading scalings,then continue to change parameter ? to get the desired prefactor.In the end,we mainly summarize the existing work and propose the future research direction.