Calculate probability of nodes in a graph I have the following graph:

A and F post the same joke on Facebook. Now there is a probability of 0.6 that node b will post the joke too. and so on...
So the weights on the edges say how likely it is that this node will post the joke if the previous node posted it.
Now I wonder how many of the nodes will have posted this joke in average after 4 Timesteps.
Can anyone help me with this? I am very new to probability-theory.
Thank you very much!
 A: I have done this the hard way with extensive tree diagrams and excel...
b is easy. Only a feeds into it (the feedback loop from c is irrelevant). So the odds of it having posted is simply 1-0.4^4 = 0.9744.
c is slightly harder. All options to get to c in 4 or less steps:
(ab and bc are edges. n = no connection)


*

*ab bc         (0.6*0.1)

*ab nbc bc     (0.6*0.9*0.1)

*ab nbc nbc bc (0.6*0.9*0.9*0.1)

*nab ab bc     (0.4*0.6*0.1)

*nab ab nbc bc (0.4*0.6*0.9*0.1)

*nab nab ab bc (0.4*0.4*0.6*0.1)


All these added up make 0.2178
h looks a bit like c's of course. g can only be reached when b, c and h posted, so it's 0.4*0.1*0.5*0.4 = 0.012.
The tree diagrams for d and e are pretty complex, because they feed from two ends. There are 53 different paths for f and d through which e can be reached. For example:


*

*ab nfd nbe nfd bd nbe nde nbe de (in which d shares b's post. e waits one time period to share that post.)

*nab fd nab nde nab de (in which d reposts f's, but b never shares. e waits one time period to share d's post.)

*nab nfd nab nfd ab fd be (in which b and d share the post after waiting 2 periods, then e reposts b's post.)


This took a lot of time and a large sheet. Unfortunately, I don't know how to do it any quicker. I suspect matrices might do the trick, but I am too unfamiliar with them.
I think the odds look like this then:


*

*a. 100%

*b. 97,4%

*c. 21,8%

*d. 86,2%

*e. 60,6%

*f. 100%

*g. 1,2%

*h. 8,4%

