Not to repeat the same thing, please just read the question in:

Constructing transition probability matrix

and the answer by @Anton Antonov and @kglr.

I like to create a transition matrix using a weighted, directed graph with a domain of weights in the interval [0, 1.5].

My states will be randomly chosen intervals. I may divide the domain into equal sub-intervals or different-length intervals at my choice. The transition matrix will consist of the number of binary links in each subset.

The following weighted graphs with 5 vertices can be used to answer.

SeedRandom[0];
matT1=RandomReal[{0,1.5}, {5,5}];  (* time t *)
matT2=RandomReal[{0,1.5}, {5,5}];  (* time t+1 *)

EDIT 1

1. Suppose that I have three states at time t: {s1, s2, s3} represented respectively by three intervals of edge weights: {r1, r2, r3} where r1=[0, 0.3], r2=(0.3, 0.6], r3=(0.6, 3] with, say, max weight is 3 in both graphs.
2. Find the number and actual linkages in each interval at t and then by using time t+1 digraph, identify the number of linkages that start from r1 at t and moving to r2 and r3 at t+1 and remaining at r1; We need to repeat the same procedure for 3 states to find out the final transition matrix.
3. Eventually, we need to derive a transition matrix using two digraphs that reflect the transition of linkages from one interval at t to another at t+1 (a typical transition matrix).
4. Additionally, I like to get the list of binary linkages themselves to analyze the set of linkages associated with a limiting probability. If the limiting probability is 0.25 for s1 then I like to know which linkages are more likely to be in that state.

I hope my explanation is clear.