  Random walks including non-nearest-neighbor jumps appear in many real
situations such as the diffusion of adatoms and have found numerous
applications including PageRank search algorithm, however, related theoretical
results are much less for this dynamical process. In this paper, we present a
study of mixed random walks in a family of fractal scale-free networks, where
both nearest-neighbor and next-nearest-neighbor jumps are included. We focus on
trapping problem in the network family, which is a particular case of random
walks with a perfect trap fixed at the central high-degree node. We derive
analytical expressions for the average trapping time (ATT), a quantitative
indicator measuring the efficiency of the trapping process, by using two
different methods, the results of which are consistent with each other.
Furthermore, we analytically determine all the eigenvalues and their
multiplicities for the fundamental matrix characterizing the dynamical process.
Our results show that although next-nearest-neighbor jumps have no effect on
the leading sacling of the trapping efficiency, they can strongly affect the
prefactor of ATT, providing insight into better understanding of random-walk
process in complex systems.
