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In this paper, we present the weighted scale-free treelike networks controlled by the weight factor  r  and the parameter  m . Based on the network structure, we study two types of weight-dependent walks with a highest-degree trap. One is standard weight-dependent walk, while the other is mixed weight-dependent walk including both nearest-neighbor and next-nearest-neighbor jumps. Although some properties have been revealed in weighted networks, studies on mixed weight-dependent walks are still less and remain a challenge. For the weighted scale-free treelike network, we derive exact solutions of the average trapping time (ATT) measuring the efficiency of the trapping process. The obtained results show that ATT is related to weight factor  r , parameter  m  and spectral dimension of the weighted network. We find that 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. Furthermore, the obtained results show that changing the walking rule has no effect on the leading scaling of the trapping efficiency. All results in this paper can help us get deeper understanding about the effect of link weight, network structure and the walking rule on the properties and functions of complex networks.

Two types of weight-dependent walks with a trap in weighted scale-free treelike networks