This document discusses different damping functions that can be used for link-based ranking algorithms like PageRank. It begins by rewriting the PageRank algorithm and defining branching contribution of a path. It then presents an explicit formula for PageRank and generalizes it to a functional ranking using different damping functions. It proposes a linear damping function and shows how to calculate rankings using this function without storing the full path matrices. Finally, it provides pseudocode for the algorithm. The key points are: rewriting PageRank, generalizing to different damping functions, proposing a linear damping, and giving an algorithm to compute rankings with the linear damping efficiently.