  In general, the power-law degree distribution has profound influence on
various dynamical processes defined on scale-free networks. In this paper, we
will show that power-law degree distribution alone does not suffice to
characterize the behavior of trapping problem on scale-free networks, which is
an integral major theme of interest for random walks in the presence of an
immobile perfect absorber. In order to achieve this goal, we study random walks
on a family of one-parameter (denoted by $q$) scale-free networks with
identical degree sequence for the full range of parameter $q$, in which a trap
is located at a fixed site. We obtain analytically or numerically the mean
first-passage time (MFPT) for the trapping issue. In the limit of large network
order (number of nodes), for the whole class of networks, the MFPT increases
asymptotically as a power-law function of network order with the exponent
obviously different for different parameter $q$, which suggests that power-law
degree distribution itself is not sufficient to characterize the scaling
behavior of MFPT for random walks, at least trapping problem, performed on
scale-free networks.
