Exploring Trapping Problem on Weighted Heterogeneous Networks Induced by Extended Corona Product

Abstract

Since many dynamical processes can be analyzed in the framework of trapping problem, it is great important to explore trapping problem on diverse complex systems or networks. However, addressing trapping problem on networks generated by graph products is still less touched. In this paper, by taking type of random walks and number of traps into account and compiling four different scenarios, trapping problem is touched more completely on resultant weighted heterogeneous networks of extended corona product and weight reinforcement mechanism. In more detail, we study standard random walk and delayed random walk on the network with a deep trap fixed at one initial node in the first and second scenarios respectively. This two types of random walks are further investigated on the network with three traps placed at initial three nodes in more challenging third and fourth scenarios. In all this four scenarios, the solutions of average trapping time( $ATT$ ) are deduced analytically to measure trapping efficiency, which agree well with their corresponding numerical counterparts and show that $ATT$ grows sub-linearly with network size. Besides, expressions of $ATT$ obtained in the second and fourth scenarios indicate that the parameter $p$ governing delayed random walk alters the pre-factor of $ATT$ but has no effect on the leading scaling of $ATT$ . Furthermore, the comparisons between expression of $ATT$ in first and third scenarios, expression of $ATT$ in second and fourth scenarios imply that $ATT$ can be lowered and trapping efficiency can be improved accordingly by introducing more traps. In summary, this work may enrich the clues for understanding trapping issue and modulating trapping process on more general heterogeneous weighted networks.

Keywords

• Random walk
• average trapping time
• heterogeneous weighted network
• corona product