Let a digraph, G, be given, where t[i,j] and a[j,i], respectively, denote transfer capacity of i allocated for transferring its information to j and absorption capacity of j allocated for absorbing information coming from i. The term, x[i], defines the information holding capacity of vertex i. If the amount of information inflowing to vertex i is larger than its capacity, then vertex i will refuse the extra information.
The following code generates an example digraph G with n=6:
Clear[trs, abs, info, edgeCapMat, system1, reducedSystem1, sa, wG1];
SeedRandom[14];
n = 6;
d = 0.3;
G1 = RandomGraph[{Round[n], Round[n*(n - 1)*d]},
DirectedEdges -> True];
trs = Table[ Table[t[i, j], {j, 1, n}], {i, 1, n}];
abs = Table[ Table[a[i, j], {j, 1, n}], {i, 1, n}] // Transpose;
info = Table[x[i], {i, 1, n}];
infoStocks[stock_] := DiagonalMatrix[stock];
edgeCapMat[trsCap_, absCap_] := (trsCap*absCap) -
DiagonalMatrix[Diagonal[trsCap*absCap]]; (* @JMissomewhatokay's contribution *)
system1 = infoStocks[info].edgeCapMat[trs, abs];
reducedSystem1 =
AdjacencyMatrix[G1]*
system1; (* the system associated with AdjacencyGraph "G" *)
sa = SparseArray[reducedSystem1];
wG1 = Graph[sa["NonzeroPositions"], EdgeWeight -> sa["NonzeroValues"],
DirectedEdges -> True, VertexLabels -> "Name" ,
EdgeLabels -> "EdgeWeight"]; (* symbolic weighted-G1 *)
(* @kglr's contribution *)
ClearAll[edgeW];
edgeW = Module[{g = #,
e = DirectedEdge @@@ Partition[#, 2, 1] & /@
FindPath[##, \[Infinity], All]},
Transpose[{e, PropertyValue[{g, #}, EdgeWeight] & /@ # & /@ e}]] &;
edgeW[wG1, 5, 2]
HighlightGraph[wG1, edgeW[wG1, 5, 2][[All, 1]]]
(Note: Mathematica Code to produce G has been developed by @kglr (many of us know @kglr from his contributions in MSE):
One can formulate a Maximum Flow problem for each path in G. Below I present an example max-flow problem from vertex 5 to vertex 2. (This is only one of the 30 potential problems with n=6.)
To understand what I am doing, one may simply follow the digraph G given above. Some of the vertices and parameters are not required for the paths from 5 to 2. For example, t[2]==0 and a[5]==0 are irrelevant by construction, and vertex 6, not involved in any of the paths concerned, should be excluded from the following specific problem formulation.
Clear[objFn, constraintsAll, choiceVars, fc1, fc2, fc3, fc4, fc5, fc6, fct1, fct2, fct3, fct4, fct5, fca1, fca2, fca3, fca4, fca5, bc1, bc2, bc3, bc4, tc1, tc2, tc3, tc4, ac1, ac2, ac3, ac4];
(* Given *)
{x[1] = 5, x[2] = 9, x[3] = 3, x[4] = 6, x[5] = 7, x[6] = 2};
{t[1] = 0.7, t[2] = 0, t[3] = 0.8, t[4] = 0.3, t[5] = 0.8, t[6] = 0.4};
{a[1] = 0.9, a[2] = 0.7, a[3] = 0.6, a[4] = 0.7, a[5] = 0, a[6] = 0.7};
Maximize ObjFn = t[5,3] a[3,5] x[5]
subject to:
(* feasibility constraints for: vertex information stock capacity *)
fc1 = t[5, 3] a[3, 5] x[5] <= x[3];
fc2 = t[3, 4] a[4, 3] x[3] <= x[4];
fc3 = t[3, 1] a[1, 3] x[3] <= x[1];
fc4 = t[4, 2] a[2, 4] x[4] <= x[2]; (* NEW: to be added *)
fc5 = t[1, 2] a[2, 1] x[1] <= x[2]; (* NEW: to be added *)
fc6 = t[4, 2] a[2, 4] x[4] + t[1, 2] a[2, 1] x[1] <= x[2]; (* NEW: to be added *)
(* feasibility constraints for: attention allocation *)
fct1 = 0 <= t[5, 3] <= 1;
fct2 = 0 <= t[3, 4] <= 1;
fct3 = 0 <= t[3, 1] <= 1;
fct4 = 0 <= t[4, 2] <= 1;
fct5 = 0 <= t[1, 2] <= 1;
fca1 = 0 <= a[3, 5] <= 1;
fca2 = 0 <= a[4, 3] <= 1;
fca3 = 0 <= a[1, 3] <= 1;
fca4 = 0 <= a[2, 4] <= 1;
fca5 = 0 <= a[2, 1] <= 1;
(* flow balancing conditions: inflow into i = outflow from i *)
bc1 = t[5, 3] a[3, 5] x[5] == (t[3, 4] a[4, 3] + t[3, 1] a[1, 3]) x[3]; (* inflow from 5 to 3 = outflow from 3 to 4 and 1 *)
bc2 = t[3, 4] a[4, 3] x[3] == t[4, 2] a[2, 4] x[4]; (* inflow from 3 to 4 = outflow from 4 to 2 *)
bc3 = t[3, 1] a[1, 3] x[3] == t[1, 2] a[2, 1] x[1]; (* NEW to be added: inflow from 3 to 1 = outflow from 1 to 2 *)
bc4 = t[4, 2] a[2, 4] x[4] + t[1, 2] a[2, 1] x[1] == t[5, 3] a[3, 5] x[5]; (* NEW to be added: total inflow to 2 = total outflow from 5 *)
(* transfer attention allocation constraints *)
tc1 = t[5, 3] <= t[5]; (* v5 allocates its total transfer attention to v3 *)
tc2 = t[3, 4] + t[3, 1] <= t[3]; (* v3 allocates its total transfer attention to v4 and v1 *)
tc3 = t[4, 2] <= t[4];
tc4 = t[1, 2] <= t[1];
(* absorption attention allocation constraints *)
ac1 = a[3, 5] <= a[3]; (* choice variable of v3 because v3 decides how much attention it should allocate for receiving information from v5 *)
ac2 = a[4, 3] <= a[4]; (* the same argument as above *)
ac3 = a[1, 3] <= a[1];
ac4 = a[2, 4] + a[2, 1] <= a[2]; (* v2 allocates its total attention b/w v4 and v1 *)
(* list of constraints *)
constraintsAll = {fc1, fc2, fc3, fc4, fc5, fc6, fct1, fct2, fct3, fct4, fct5, fca1, fca2, fca3, fca4, fca5, bc1, bc2, bc3, bc4, tc1, tc2, tc3, tc4, ac1, ac2, ac3, ac4};
(* list of choice variables: find the optimal values of these vars *)
choiceVars = {t[5, 3], t[3, 4], t[3, 1], t[4, 2], t[1, 2], a[3, 5],
a[4, 3], a[1, 3], a[2, 4], a[2, 1]}
Maximize[{objFn, constraintsAll}, choiceVars]
The solution:
{2.16, {t[5, 3] -> 0.786664, t[3, 4] -> 0., t[3, 1] -> 0.8, t[4, 2]->0.278239,
t[1, 2] -> 0.671632, a[3, 5] -> 0.392253, a[4, 3] -> 4.26136*10^-10,
a[1, 3] -> 0.9, a[2, 4] -> 0., a[2, 1] -> 0.643209}}
Note: The above max-flow is only for the paths from 5 to 2. I want to solve the same problem for each one of the existing paths in G, and each problem should give me a set of values for {t[i,j], a[j,i]} that maximize the relevant objective function. Needless to say, each problem will have its own specific constraints and objective function to maximize.
Question
How can I automate the above problem formulation to find the system-wide optimal values for each path, i.e., {t[i,j], a[j,i] for all i and j}?.
(Note: This question is a simplified version of a question asked in MSE before. I made some improvement in the formulation.)
a[6],t[6]andx[6]really needed as part of the formulation? Or can we just restrict attention to the subgraphs associated with the paths between node 5 and node 2? – kglr Oct 08 '18 at 03:48x[3] == 2, andx[4] == 1, the proposed solution violates constraintbc2. Checksoln = {t[5, 3] -> 0.8, t[3, 4] -> 0.31903, t[3, 1] -> 0.48097, t[4, 2] -> 0.3, t[1, 2] -> 0.7, a[3, 5] -> 0.6, a[4, 3] -> 0.7, a[1, 3] -> 0.9, a[2, 4] -> 0., a[2, 1] -> 0.7}; bc2 = t[3, 4] a[4, 3] x[3] == t[4, 2] a[2, 4] x[4]; bc2 /. {x[3] -> 2, x[4] -> 1} /. soln– kglr Oct 08 '18 at 04:55bc123rewritesbc1assumingbc2andbc3are satisfied; but this does not mean you can eliminatebc2andbc3. You still need the flow balance at nodes 4 and 2. If you addbc2andbc3to the constraint set, the proposed solution still violatesbc2. Imho, easier way to get around infeasibilities is to change the attention allocation constraints to inequalities (allowing some part of attention budgets to be unspent). – kglr Oct 08 '18 at 11:18