Quantifying the effect of nonnearest-neighbor hop on transport efficiency in the presence of reciprocity

Nonnearest-neighbor hop during random walks is observed in various real situations and their nontrivial roles are also confirmed. On the other hand, the reciprocity also can significantly impact dynamical processes on weighted networks. However, the roles of nonnearest-neighbor hop are less explored in the presence of reciprocity. Here, we quantify the effect of nonnearest-neighbor hop on random walks in the presence of reciprocity. Specifically, a class of directed weighted fractal scale-free networks is proposed to accommodate reciprocity. Subsequently, mixed random walk that includes both nearest-neighbor and next-nearest-neighbor jump is defined on the networks. Finally, a trap is placed at a node of the networks and the average trapping time (ATT) is derived analytically and exactly to determine the effect of next-nearest-neighbor hop on transport efficiency. The obtained closed form solution of ATT shows that (1) Dependent on the combination of values of parameters governing network evolution and reciprocity, ATT can grow linearly, sub-linearly or super-linearly with network size. (2) The parameter manipulating mixed random walk only modifies the prefactor of ATT, so the next-nearest-neighbor hop has no effect on the scaling of transport efficiency in the presence of reciprocity. (3) The weight parameter enters into both the prefactor and leading scaling of ATT, so reciprocity impacts the transport efficiency both qualitatively and quantitatively.