  We develop a generalized optimization framework for graph-based
semi-supervised learning. The framework gives as particular cases the Standard
Laplacian, Normalized Laplacian and PageRank based methods. We have also
provided new probabilistic interpretation based on random walks and
characterized the limiting behaviour of the methods. The random walk based
interpretation allows us to explain di erences between the performances of
methods with di erent smoothing kernels. It appears that the PageRank based
method is robust with respect to the choice of the regularization parameter and
the labelled data. We illustrate our theoretical results with two realistic
datasets, characterizing di erent challenges: Les Miserables characters social
network and Wikipedia hyper-link graph. The graph-based semi-supervised
learning classi- es the Wikipedia articles with very good precision and perfect
recall employing only the information about the hyper-text links.
