Conventional wisdom is that the power-law degree sequence has a profound impact on diverse dynamical processes taking place on scale-free networks. Here we show that power-law degree distribution alone is not sufficient to determine the behaviors of coherent exciton transport on scale-free networks. For this purpose, we study coherent exciton transport on a class of scale-free networks controlled by a tunable parameter q, which have the same degree sequence independent of q, but are very different in other structural properties determined by q, such as average distance and fractality. First, through numerical determination of the time evolutions of return probabilities, we compare in detail the dissimilarities in oscillation periods and oscillation amplitudes between two limiting cases of q = 0 and q = 1. Second, we demonstrate that, in the network of q = 0 up to the third generation, the quantum transport problem between node groups, partitioned by node degrees, can be mapped onto a line. Furthermore, we show that in the case of long time limit, the transition probabilities in the networks of q = 0 and q = 1 exhibit different characteristic patterns with identical values. Finally, we find that the degree-averaged return probability decreases when the parameter q increases from 0 to 1, indicating that the stronger localization can be found in those networks with lower values of q. Our results suggest that it is far from enough to characterize the behaviors of coherent exciton transport on scale-free networks merely by power-law degree distribution.
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