How to convert a weighted, directed graph into a discrete Markov transition matrix

Question

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 five states s1=[0, 0.5], s2=(0.5, 1], s3=(1, 1.5] etc.

r1T1=BoolEval[0<= matT1<=0.5]/.{1->s1};
r2T1=BoolEval[0.5<matT1<= 1]/.{1-> s2};
r3T1=BoolEval[1<matT1<=1.5]/.{1 -> s3};
r4T1=BoolEval[1.5<matT1<=2]/.{1 -> s4};
r5T1=BoolEval[2<matT1<=2.5]/.{1 -> s5};
matT1S = r1T1 + r2T1 + r3T1 + r4T1 + r5T1 // 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};
r4T2=BoolEval[1.5<matT2<=2]/.{1 -> s4};
r5T2=BoolEval[2<matT2<=2.5]/.{1 -> s5};
matT2S = r1T2 + r2T2 + r3T2 + r4T2 + r5T2 // 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 matT2Sby manually comparing the states in both matrices.

Clear[n, states, map];
n = Length[matT2S];
states = {s1, s2, s3, s4, s5};
map = {};

Do[
   If[matT1S[[i, j]] == states[[1]] &&  
      matT2S[[i, j]] == states[[2]], 
      AppendTo[map, {i, j}]
     ], {i, n}, {j, n}
  ]  
 
Length[map]   (* gives 0 *)

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

Rows are associated with time t and columns with t+1. This map illustrates that, out of 3 links in state s1 at time t, 1 remains in s1 at t+1, and 1 moves to s3 at t+1 and 1 moves to s5 at t+1. Other numbers in the map should be read likewise. Using this map,

traMap={
         {1,0,1,0,1},
         {1,0,1,1,0},
         {0,2,0,0,0}, 
         {1,1,2,4,2},
         {0,1,2,3,1}
       };
transMatrix=
  DiagonalMatrix[1/Total[traMap,
  {2}]].traMap

A row-stochastic transition matrix as:

transMatrix = {
    {1/3,   0,    1/3,   0,   1/3},
    {1/3,   0,    1/3,  1/3,   0 },
    {0,     1,     0,    0,    0 },
    {1/10, 1/10,  1/5,  2/5,  1/5},
    {0,    1/7,   2/7,  3/7,  1/7}
  };

and

MatrixPower[transMatrix, 100] 

produces the following limiting distribution:

This limiting distribution translates the current vector (3, 3, 2, 10, 7) to (0.17, 0.26, 0.22, 0.23, 0.12)*(3, 3, 2, 10, 7).

My question: Although I found out the transition, I do not know which linkages are in each state in the final period t+100. I like to know the specific linkages associated the new distribution (0.17, 0.26, 0.22, 0.23, 0.12)*(3, 3, 2, 10, 7).

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?

Answer 1

A more streamlined way to produce the matrices in OP (not an answer):

ClearAll[toStates]
toStates[t_, s_, m_] := Map[s[[Total[1 - UnitStep[t - #]]]] &, m, {2}]

thresholds = {0, .5, 1., 1.5};
states = {s1, s2, s3};

m1S = toStates[thresholds, states, matT1];
m2S = toStates[thresholds, states, matT2];

{m1S, m2S} == {matT1S, matT2S}

 True

maP[m1_, m2_, st_] := PadRight[KeySort[GroupBy[
      Join @@ (Transpose /@ Transpose[{m1, m2}]), 
      First -> Last, KeySort[Counts[#]] /@ st /. _Missing -> 0 &]] /@ 
    st /. _Missing -> {0}]

maP[m1S, m2S, states]

TeXForm @ MatrixForm @ %

$\left( \begin{array}{ccc} 2 & 3 & 1 \\ 3 & 5 & 2 \\ 2 & 4 & 3 \\ \end{array} \right)$

tm = Normalize[#, Total] & /@ map;
TeXForm @ MatrixForm @ tm

$\left( \begin{array}{ccc} \frac{1}{3} & \frac{1}{2} & \frac{1}{6} \\ \frac{3}{10} & \frac{1}{2} & \frac{1}{5} \\ \frac{2}{9} & \frac{4}{9} & \frac{1}{3} \\ \end{array} \right)$

Answer 2

I am not sure I understand the question. It seems to me that you can make the transition matrix by making the graph adjacency matrix row stochastic.

Here is an example with question’s matT1:

matT1rs = DiagonalMatrix[1/Total[matT1, {2}]].matT1;

Total[matT1rs, {2}]

(* {1., 1., 1., 1., 1.} *)

MarkovProcessProperties[DiscreteMarkovProcess[{1, 0, 0, 0, 0}, matT1rs]]