Trapping time of weighted-dependent walks depending on the weight factor

• Trapping problem for weighted-dependent walks on weighted hierarchical networks.
• Average trapping time grows differently with the network order.
• Transport in un-weighted hierarchical networks is the most efficient.

A weighted hierarchical network model is introduced in this paper. We study the trapping problem for weighted-dependent walks taking place on a hierarchical weighted network at a given trap. We concentrate on the average trapping time (ATT) for three cases, i.e., the immobile trap located at the root node, the external nodes and a neighbor of the root with a single connectivity, respectively. The closed-form formulae for the ATT for the three cases are obtained. In different range of the weight factor r, the leading term of ATT grows differently, i.e., superlinearly, linearly and sublinearly with the network size. For all the three cases of trapping problems, the leading scaling of ATT can reach the minimum scaling.