This work extends the personalized PageRank model invented by Brin and Page to a family of PageRank models with various damping schemes. The goal with increased model variety is to capture or recognize a larger number of types of network activities, phenomena and propagation patterns. The response in PageRank distribution to variation in damping mechanism is then characterized analytically, and further estimated quantitatively on 6 large real-world link graphs. The study leads to new observation and empirical findings. It is found that the difference in the pattern of PageRank vector responding to parameter variation by each model among the 6 graphs is relatively smaller than the difference among 3 particular models used in the study on each of the graphs. This suggests the utility of model variety for differentiating network activities and propagation patterns. The quantitative analysis of the damping mechanisms over multiple damping models and parameters is facilitated by a highly efficient algorithm, which calculates all PageRank vectors at once via a commonly shared, spectrally invariant subspace. The spectral space is found to be of low dimension for each of the real-world graphs.