  It has been recently reported that the reciprocity of real-life weighted
networks is very pronounced, however its impact on dynamical processes is
poorly understood. In this paper, we study random walks in a scale-free
directed weighted network with a trap at the central hub node, where the weight
of each directed edge is dominated by a parameter controlling the extent of
network reciprocity. We derive two expressions for the mean first passage time
(MFPT) to the trap, by using two different techniques, the results of which
agree well with each other. We also analytically determine all the eigenvalues
as well as their multiplicities for the fundamental matrix of the dynamical
process, and show that the largest eigenvalue has an identical dominant scaling
as that of the MFPT.We find that the weight parameter has a substantial effect
on the MFPT, which behaves as a power-law function of the system size with the
power exponent dependent on the parameter, signaling the crucial role of
reciprocity in random walks occurring in weighted networks.
