  A wide variety of real-life networks share two remarkable generic topological
properties: scale-free behavior and modular organization, and it is natural and
important to study how these two features affect the dynamical processes taking
place on such networks. In this paper, we investigate a simple stochastic
process--trapping problem, a random walk with a perfect trap fixed at a given
location, performed on a family of hierarchical networks that exhibit
simultaneously striking scale-free and modular structure. We focus on a
particular case with the immobile trap positioned at the hub node having the
largest degree. Using a method based on generating functions, we determine
explicitly the mean first-passage time (MFPT) for the trapping problem, which
is the mean of the node-to-trap first-passage time over the entire network. The
exact expression for the MFPT is calculated through the recurrence relations
derived from the special construction of the hierarchical networks. The
obtained rigorous formula corroborated by extensive direct numerical
calculations exhibits that the MFPT grows algebraically with the network order.
Concretely, the MFPT increases as a power-law function of the number of nodes
with the exponent much less than 1. We demonstrate that the hierarchical
networks under consideration have more efficient structure for transport by
diffusion in contrast with other analytically soluble media including some
previously studied scale-free networks. We argue that the scale-free and
modular topologies are responsible for the high efficiency of the trapping
process on the hierarchical networks.
