Clear[n, m, mFull, G, a, A1, A2, A3, A4, constraintsFull, reducedSystem];
SeedRandom[13];
n = 4;
m = 6;

(* Full system *)
mFull = {{0, a1[1, 2], a1[1, 3], a1[1, 4]}, {a2[2, 1], 0, a2[2, 3],
    a2[2, 4]}, {a3[3, 1], a3[3, 2], 0, a3[3, 4]}, {a4[4, 1], a4[4, 2],
     a4[4, 3], 0}};

(* constraints for individual elements in "mFull" *)
constraintsFull = Table[{
    0 <= a1[1, j] <= 1,
    0 <= a2[2, j] <= 1,
    0 <= a3[3, j] <= 1,
    0 <= a4[4, j] <= 1}, {j, 1, n}];  (* for all j *)

(* define Upper Limits for "mFull" *)
upperLimitFull = {
   {A1Full = Total[Ta[![enter image description here][1]][1]ble[a1[1, j], {j, 1, n}] /. a1[1, 1] -> 0]},
   {A2Full = Total[Table[a2[2, j], {j, 1, n}] /. a2[2, 2] -> 0]},
   {A3Full = Total[Table[a3[3, j], {j, 1, n}] /. a3[3, 3] -> 0]},
   {A4Full = Total[Table[a4[4, j], {j, 1, n}] /. a4[4, 4] -> 0]}
   };

(* For the reduced system *)
G = RandomGraph[{n, m}, DirectedEdges -> True];
reducedSystem = AdjacencyMatrix[G]*mFull;

constraintsReduced = {
   0 <= a1[1, 4] <= 1,
    0 <= a2[2, 1] <= 1,
   0 <= a2[2, 3] <= 1,
   0 <= a3[3, 2] <= 1,
   0 <= a4[4, 1] <= 1,
   0 <= a4[4, 3] <= 1
    };

(* define Upper Limits for "reducedSystem" *)
upperLimitReduced = {
   {A1reduced = reducedSystem[[1]] // Total},
   {A2reduced = reducedSystem[[2]] // Total},
   {A3reduced = reducedSystem[[3]] // Total},
   {A4reduced = reducedSystem[[4]] // Total}
   };