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This paper introduces a novel framework, HodgeRank on Random Graphs, based on paired comparison, for subjective video quality assessment. Two types of random graph models are studied, i.e., Erdös-Rényi random graphs and random regular graphs. Hodge decomposition of paired comparison data may derive, from incomplete and imbalanced data, quality scores of videos and inconsistency of participants' judgments. We demonstrate the effectiveness of the proposed framework on LIVE video database. Both of the two random designs are promising sampling methods without jeopardizing the accuracy of the results. In particular, due to balanced sampling, random regular graphs may achieve better performances when sampling rates are small. However, when the number of videos is large or when sampling rates are large, their performances are so close that Erdös-Rényi random graphs, as the simplest independent and identically distributed sampling scheme, could provide good approximations to random regular graphs, as a dependent sampling scheme. In contrast to the traditional deterministic incomplete block designs, our random design is not only suitable for traditional laboratory studies, but also for crowdsourcing experiments on Internet where the raters are distributive and it is hard to control with fixed designs.