  The explicit determinations of the mean first-passage time (MFPT) for
trapping problem are limited to some simple structure, e.g., regular lattices
and regular geometrical fractals, and determining MFPT for random walks on
other media, especially complex real networks, is a theoretical challenge. In
this paper, we investigate a simple random walk on the the pseudofractal
scale-free web (PSFW) with a perfect trap located at a node with the highest
degree, which simultaneously exhibits the remarkable scale-free and small-world
properties observed in real networks. We obtain the exact solution for the MFPT
that is calculated through the recurrence relations derived from the structure
of PSFW. The rigorous solution exhibits that the MFPT approximately increases
as a power-law function of the number of nodes, with the exponent less than 1.
We confirm the closed-form solution by direct numerical calculations. We show
that the structure of PSFW can improve the efficiency of transport by
diffusion, compared with some other structure, such as regular lattices,
Sierpinski fractals, and T-graph. The analytical method can be applied to other
deterministic networks, making the accurate computation of MFPT possible.
