Is there any PageRank-like method on weighted graph?
Weighted versions of Pagerank do exist, and it is easy to incorporate edge weights into the PageRank algorithm (just multiply each edge's probability by its weight vector, and then normalize to make the edge probabilities to add up to 1)

The difficult question is how to set these weights. Backstrom and Leskovec propose an algorithm to learn these weights in their WSDM paper on Supervised Random Walks which they propose for use in link prediction and link recommendation.