The Average Weighted Receiving Time With Weight-Dependent Walk On A Family Of Double-Weighted Polymer Networks

In this paper, a family of the double-weighted polymer networks is introduced depending on the number of copies f and two weight factors w, r. The double-weights represent the selected weights and the consumed weights, respectively. Denote by w(ij)(S) the selected weight connecting the nodes i and j, and denote by w(ij)(C) the consumed weight connecting the nodes i and j. Let w(ij)(S )be related to the weight factor w, and let w(ij)(C) be related to the weight factors r. 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 N-g, the number of copies f , and two weight factors w, r. 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.