Is there any PageRank-like method on weighted graph? I found that PageRank algorithm depend heavily on the existence of edges, but the weights of edges give small effect. 
For example, PageRank on triangle graph(A -> B,  B -> C,  C -> A) is always (1/3, 1/3, 1/3) even if one of the weight is 0.0000001.
Is there any PageRank-like method work well on the weighted graph?

Thanks in advance!
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.