How do I combine multiple matrices into a single adjacency graph?

Question

I'm sort of new to Mathematica and have been working on a project that involves simulating newtwork changes in the form of matrices. I have the following lines of code:

k = 10;
m = 10;
m1 = SparseArray[_ :> RandomInteger[1], {k, m}];
A = UpperTriangularize[m1] + Transpose[UpperTriangularize[m1, 1]];
MatrixForm[A];
aa = MatrixForm[A, 
  TableHeadings -> {{"A1", "A2", "A3", "A4", "A5", "V1", "V2", "V3", 
     "V4", "V5"}, {"A1", "A2", "A3", "A4", "A5", "V1", "V2", "V3", 
     "V4", "V5"}}]

This creates a "randomly" generated matrix consisting of elements 1 or 0.
I have included a picture to help explain my question. It's just a randomly generated matrix from the code above but it would be easier to see visualize the next part if I included colors.

I'm trying to create a SINGLE adjacency graph which shows the relationships between A-A (blue), A-V (pink), and V-V (red). The V-A connection shown in black is just the transpose of the pink quadrant and is unnecessary for me to show. The headings of A and V just represent different molecules. The matrix elements indicate where there is an edge between two molecules. A 1 means there is an edge, a 0 means no edge. As an example A1-A1 has a 0 as its element, therefore there is no edge.

What I managed to do so far is create two adjacency matrices, one for the A-A interactions and one of the V-V interactions. That is represented by the code below. The quadrants are in reference to the same found in a Cartesian coordinate graph (Top right = 1, then go counter clockwise for the other quadrants).

(*Pulls out the submatrix in Q1 *)
sm1 = A[[1 ;; 5, 6 ;; 10]];
ns1 = Normal[sm1]

(*Pull out the submatrix in Q2*)
sm2 = A[[1 ;; 5, 1 ;; 5]];
ns2 = Normal[sm2]

(*Pulls out the submatrix in Q4*)
sm3 = A[[6 ;; 10, 6 ;; 10]];
ns3 = Normal[sm3]

(*Vertex Labels*)
vlabel2 = {A1, A2, A3, A4, A5};
vlabel3 = {V1, V2, V3, V4, V5};

{g2, g3} = (AdjacencyGraph @@@ {{vlabel2, ns2}, {vlabel3, ns3}})
Row[Labeled[
    SetProperty[#, {VertexShapeFunction -> "Name", 
      ImageSize -> 200}], #2, Top] & @@@ {{g2, "g2"}, {g3, "g3"}}]

The above code generates two adjacency matrices, one for the A-A interactions (blue region) and one for the V-V interactions (red region). In included how to pull out the submatrix for Q1 since I think that would be necessary in helping me solve my problem. The main issue I'm having is incorporating the V-A interactions (pink region). I'm lost as to how I would go about incorporating the third adjacency matrix in order to connect the two that I currently have.

Any help would be greatly appreciate!

Answer 1

Maybe something like:

k = 10;
m = 10;
SeedRandom[1]
m1 = SparseArray@RandomInteger[1, {k, m}];
A = UpperTriangularize[m1] + Transpose[UpperTriangularize[m1, 1]];
labels = {"A1", "A2", "A3", "A4", "A5", "V1", "V2", "V3", "V4", "V5"};

To get the four submatrices from A you can use Partition and to get A from the four matrices you can use ArrayFlatten:

partitionedA = {{AA, AV}, {VA, VV}} = Partition[A, {5, 5}];

Row[Riffle[MatrixForm /@ {A, partitionedA, ArrayFlatten@partitionedA}, 
   {RawBoxes @ StyleBox[UnderoverscriptBox["\[LongRightArrow]", "", 
      RowBox[{"   ", "Partition", "   "}]], 20], 
    RawBoxes @ StyleBox[UnderoverscriptBox["\[LongRightArrow]", "", 
      RowBox[{"   ", "ArrayFlatten", "   "}]], 20]}], Spacer[10]]

You can use labels as the first argument and A as the second argument to AdjacencyGraph and style each edge based on the matrix blocks its endpoints belong:

style = MapThread[Map[Function[x, Style[x, #2]], #, {-1}] &, {##},  2] &;
styledAM = MatrixForm[ArrayFlatten@ 
    style[partitionedA, {{Blue, Magenta}, {Black, Red}}], 
   TableHeadings -> {labels, labels}];

ag = AdjacencyGraph[labels, A, DirectedEdges -> True, 
   GraphLayout -> {"MultipartiteEmbedding", "VertexPartition" -> {5, 5}},
    VertexLabels -> Placed["Name", Center], VertexSize -> Large, 
   EdgeStyle -> {e_ :>  Switch[StringTake[List @@ e, 1], 
     {"A", "A"}, Blue, {"V", "V"}, Red, {"A", "V"}, Magenta, _, Black]}, 
   ImageSize -> Large];

Row[{ag, styledAM}]

You can select the edges associated with each matrix block and construct a separate graph for each block:

colors = AssociationThread[{"A - A", "A - V", "V - V", "V - A"}, 
   PropertyValue[{ag, #}, EdgeStyle] & /@ 
 {"A1" \[DirectedEdge] "A1", "A1" \[DirectedEdge] "V3", 
  "V1" \[DirectedEdge] "V1", "V1" \[DirectedEdge] "A5"}];

{gAA, gAV, gVA, gVV} = Graph[Select[EdgeList[ag], 
      Function[e, PropertyValue[{ag, e}, EdgeStyle] == colors@#]], 
     EdgeStyle -> colors@#, ImageSize -> Medium, 
     VertexLabels -> Placed["Name", Center], VertexSize -> Large, 
     VertexCoordinates -> {v_ :> vCoords[v]}] & /@ Keys[colors];

Row@MapThread[Labeled[##, Top] &, {{gAA, gAV, gVA, gVV}, Style[#, 16] & /@ Keys[colors]}] 

Finally, to get from graphs gAA, gAV, gVA and gVV to the combined graph ag you can use GraphUnion:

GraphUnion[gAA, gAV, gVA, gVV, ## & @@ Options[ag]]

Answer 2

You do not need to disassemble the adjacency matrix. You can assign colour within a single graph.

names = {"A1", "A2", "A3", "A4", "A5", "V1", "V2", "V3", "V4", "V5"};

type = StringTake[#, 1] &; (* what type of node? A or V? *)

colorRules = {
   {"A", "A"} -> Blue,
   {"V", "V"} -> Red,
   {"A", "V"} -> Purple
   };

AdjacencyGraph[{"A1","A2","A3","A4","A5","V1","V2","V3","V4","V5"},A,
    EdgeStyle -> { Thick, UndirectedEdge[u_, v_] :> Replace[Sort@{type[u],type[v]}, colorRules] },
    GraphStyle -> "IndexLabeled" (* this is a misnomer--it labels by name, not index *)
]