Intuitively, link weight could affect the dynamics of the network. However, the theoretical research on the effects of link weight on network dynamics is still rare. In this paper, we present two heterogeneous weighted pseudo-fractal webs controlled by two weight parameters [Formula: see text] and [Formula: see text] ([Formula: see text]). Both graph models are scale-free deterministic graphs, and they have the same weight sequence when [Formula: see text] and [Formula: see text] are fixed. Based on their self-similar graph structure, we study the effect of heterogeneous weight on the random walks in graph with scale-free characteristics. We obtain analytically the average trapping time (ATT) for biased random walks in graphs with a trap located at a fixed node. Analyzing and comparing the obtained solutions, we find that in the large graph limit, the ATT for both graph models all grow as a power function of the graph size (number of nodes) with the exponent [Formula: see text] dependents on the ratio of parameters [Formula: see text] and [Formula: see text], but their exponents [Formula: see text] are not the same, one gets the minimum when [Formula: see text], while the other gets the maximum. Furthermore, the average weighted shortest path length (AWSPL) to the trap is calculated for both graph models, respectively. We show that when the graph size tends to infinity, their AWSPL grows unbounded with the graph size for most parameters. We hope that these results could help people understand the impact of heterogeneous weight on network dynamics.
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