Suppose that I have matrix `matT1` at time t and `matT2` at time t+1:

    matT1 = {
             {0.98, 0.95, 1.00, 0.85, 1.40}, 
             {1.46, 0.36, 0.96, 0.15, 0.97}, 
             {0.24, 1.20, 1.40, 0.96, 0.46}, 
             {1.10, 1.30, 0.03, 0.81, 0.53}, 
             {1.30, 1.50, 1.30, 0.51, 0.42}
           };

    matT2 = {
             {0.44, 1.00, 0.77, 1.20, 0.61}, 
             {0.58, 0.57, 0.65, 0.19, 1.00}, 
             {1.40, 0.14, 1.20, 1.40, 0.96}, 
             {1.40, 0.95, 0.74, 0.56, 0.47}, 
             {0.98, 0.45, 1.30, 0.34, 0.25}
           };

Note that these matrices represent two different weighted directed graphs with 5 vertices. Elements of these two matrices are assigned to one of the three states `s1=[0, 0.5]`, `s2=(0.5, 1]`, and `s3=(1, 1.5]`. 

    r1T1=BoolEval[0<= matT1<=0.5]/.{1->s1};
    r2T1=BoolEval[0.5<matT1<= 1]/.{1-> s2};
    r3T1=BoolEval[1<matT1<=1.5]/.{1 -> s3};
    matT1S = r1T1 + r2T1 + r3T1 // MatrixForm

    r1T2=BoolEval[0<=matT2<=0.5]/.{1 -> s1};
    r2T2=BoolEval[0.5<matT2<=1]/.{1 -> s2};
    r3T2=BoolEval[1<matT2<=1.5]/.{1 -> s3};
    matT2S = r1T2 + r2T2 + r3T2 // MatrixForm

respectively yield:
  
    matT1S = {
              {s2, s2, s2, s2, s3},
              {s3, s1, s2, s1, s2},
              {s1, s3, s3, s2, s1},
              {s3, s3, s1, s2, s2},
              {s3, s3, s3, s2, s1}
             };
    matT2S = {
              {s1, s2, s2, s3, s2},
              {s2, s2, s2, s1, s2},
              {s3, s1, s3, s3, s2},
              {s3, s2, s2, s2, s1},
              {s2, s1, s3, s1, s1}
             };
  
We then derive a map of transition from `matT1S` to `matT2S`by manually comparing the states in both matrices. 

    Clear[n, states, map];
    n = Length[matT2S];
    states = {s1, s2, s3};
    map = {};

    Do[
       If[matT1S[[i, j]] == states[[1]] &&  
          matT2S[[i, j]] == states[[2]], 
          AppendTo[map, {i, j}]
         ], {i, n}, {j, n}
      ]  (* states[[i]] index should be changed for each transition type. This code generates the number of transition (3) from `s1` to `s2` only.*)
     
    Length[map]   (* gives 3 *)

For each pair of `states`, I run the above code to obtain the following map:

[![enter image description here][1]][1]

Rows are associated with time t and columns  with t+1. This map illustrates that, out of 6 links in state `s1` at time t, 2 remain in `s1` at t+1, and 3 move to `s2` at t+1 and 1 moves to `s3` at t+1. Other numbers in the map should be read likewise. Using this map, we calculate a row-stochastic transition matrix as:

    transMatrix = {
                   {2/6,  3/6,  1/6}, 
                   {3/10, 5/10, 2/10},
                   {2/9,  4/9,  3/9}
                 };
  
produces
    
[![enter image description here][2]][2]
  
and 

    MatrixPower[transMatrix, 100] 

produces the following limiting distribution: 

[![enter image description here][3]][3]

This limiting distribution translates the current vector `(6, 10, 9)` to `(0.29, 0.49, 0.22)*(6, 10, 9) = (7,25, 12.25, 5.5)`. This means that states `s1` and `s2` host more links while state `s3` looses its members. 

My question: Although I found out the change shown as `(7,25, 12.25, 5.5)`, I do not know which linkages are in each state in the final period t+100. I know that 7.25 will be in state `s1` after 100 repetition of transition but which linkages are they? Basically, I like to know the specific linkages associated with `(7.25, 12.25, 5.5)`.

Would it be possible to write a function `transMatrix[matrixT_,matrixT1_]:=...` to produce: a transition matrix (row stochastic matrix), a final distribution of the linkages across three states, and subsets of the linkages across each state?

  [1]: https://i.sstatic.net/HM3N4.png
  [2]: https://i.sstatic.net/kAKkz.png
  [3]: https://i.sstatic.net/SoT25.png