Calculations of first passage time of delayed tree-like networks

In this paper, we study random walks in a family of delayed tree-like networks controlled by two network parameters, where an immobile trap is located at the initial node. The novel feature of this family of networks is that the existing nodes have a time delay to give birth to new nodes. By the self-similar network structure, we obtain exact solutions of three types of first passage time (FPT) measuring the efficiency of random walks, which includes the mean receiving time (MRT), mean sending time (MST) and mean first passage time (MFPT). The obtained results show that the MRT, MST and MFPT increase with the network parameters. We further show that the values of MRT, MST and MFPT are much shorter than the nondelayed counterpart, implying that the efficiency of random walks in delayed trees is much higher.