The average weighted receiving time with weight-dependent walk on a family of double-weighted polymer networks

Abstract

In this paper, a family of the double-weighted polymer networks is introduced depending on the number of copies [Formula: see text] and two weight factors [Formula: see text]. The double-weights represent the selected weights and the consumed weights, respectively. Denote by [Formula: see text] the selected weight connecting the nodes [Formula: see text] and [Formula: see text], and denote by [Formula: see text] the consumed weight connecting the nodes [Formula: see text] and [Formula: see text]. Let [Formula: see text] be related to the weight factor [Formula: see text], and let [Formula: see text] be related to the weight factors [Formula: see text]. Assuming that the walker, at each step, starting from its current node, moves to any of its neighbors with probability proportional to the selected weight of edge linking them. The weighted time for two adjacency nodes is the consumed weight connecting the two nodes. The average weighted receiving time (AWRT) is defined on the double-weighted polymer networks. Our results show that in large network, the leading behaviors of AWRT for the double-weighted polymer networks follow distinct scalings, with the trapping efficiency associated with the network size [Formula: see text], the number of copies [Formula: see text], and two weight factors [Formula: see text]. We also found that the scalings of the AWRT with weight-dependent walk in double-weighted polymer networks is due to the use of the weight-dependent walk and the weighted time. The dominant reason is the range of each weight factor. To investigate the reason of the scalings, the AWRT for four cases are discussed.