  Random walks constitute a fundamental mechanism for a large set of dynamics
taking place on networks. In this article, we study random walks on weighted
networks with an arbitrary degree distribution, where the weight of an edge
between two nodes has a tunable parameter. By using the spectral graph theory,
we derive analytical expressions for the stationary distribution, mean
first-passage time (MFPT), average trapping time (ATT), and lower bound of the
ATT, which is defined as the average MFPT to a given node over every starting
point chosen from the stationary distribution. All these results depend on the
weight parameter, indicating a significant role of network weights on random
walks. For the case of uncorrelated networks, we provide explicit formulas for
the stationary distribution as well as ATT. Particularly, for uncorrelated
scale-free networks, when the target is placed on a node with the highest
degree, we show that ATT can display various scalings of network size,
depending also on the same parameter. Our findings could pave a way to
delicately controlling random-walk dynamics on complex networks.
