There is an analogy between the study of metric graphs and the study of Riemann surfaces, and a question is to construct uniformization theorem for metric graphs which would require a concept of "hyperbolic metric" on it. With Farbod Shokrieh, we found a graph theoretic analogy of a classical result by Kazhdan on the limit of canonical, or Bergman metric under a tower of normal covers, which indicates that the limiting metric might be such a candidate. I will also discuss generalizations of it to higher dimensional simplicial complex and some further questions.