probability of each terminal node in a directed graph

Question

I have a directed graph G(V,E) and weight w(u,v).

In this graph weight w(u,v) represents how many times the node(v) has visited from node(u). for example(See this for a directed graph image):

     1        3
  A ----- B ----- D
  | \____/|
 1|   4   |2
  |       |
  C       E

As C and B are visited once from A, D is visited 3 times from B and so on. Given this data how can I calculate exact probability to reach each terminal node i.e; C,E,D, if starting from A.

Any suggestion?

Answer 1

The following are the un-normalized and then row-normalized transition matrices of the markov chain, also shown in the figure. We need to calculate the absorption probabilities as shown in the figure.

  A B C D E
A 0 1 1 0 0
B 4 0 0 3 2
C 0 0 0 0 0
D 0 0 0 0 0
E 0 0 0 0 0  

          A   B   C         D         E
A 0.0000000 0.5 0.5 0.0000000 0.0000000
B 0.4444444 0.0 0.0 0.3333333 0.2222222
C 0.0000000 0.0 0.0 0.0000000 0.0000000
D 0.0000000 0.0 0.0 0.0000000 0.0000000
E 0.0000000 0.0 0.0 0.0000000 0.0000000