Creating a directed weighted graph by using a database

Question

Given a database in matrix format:

SeedRandom[6];
mat = RandomInteger[5, {27, 30}];
mat[[1, All]] = {"", a1, a2, a3, a4, a5, b1, b2, b3, b4, b5, c1, c2, 
   c3, c4, c5, d1, d2, d3, d4, d5, afd1, afd2, bfd1, bfd2, cfd1, cfd2,
   dfd1, dfd2, TD};
mat[[All, 1]] = {"", a1, a2, a3, a4, a5, b1, b2, b3, b4, b5, c1, c2, 
   c3, c4, c5, d1, d2, d3, d4, d5, aT, bT, cT, dT, VA, TS};

Each row and column has a name, e.g., a1, a2, .... Suppose the following operations on mat.

1. Drop the columns {a1, a3, b1, b2, c4, c5, d2, d5} and the rows with the same names.
2. Create a weighted directed graph using the columns {a2, a4, a5, b3, b4, b5, c1, c2, c3, d1, d3, d4} and the rows with the same names.

My question is not about the operations referred to in 1 and 2. I like to know how to keep the linkage between a string vertex name and a numeric vertex name. String names for the list in 2 are {a2, a4, a5, b3, b4, b5, c1, c2, c3, d1, d3, d4} and the associated numeric names are {2, 4, 5, 8, 9, 10, 11, 12, 13, 16, 18, 19}.

Because my original matrix is large, with the operation in item 1, I lose the linkage between numeric and string names, which I need in later stages of output formatting. For example, I can easily find Cliques with numeric vertex names but I need to know the string names linked to the numeric vertices.

Any suggestion?

EDIT 1

Suppose that

mat1 = Drop[Take[mat,21,21], {3,15},{3,15}];
MatrixForm@mat1

Using mat1, a directed graph given below is constructed by:

select[matrix_, lB_, uB_] := 
  matrix*Map[Boole[lB <= # <= uB] &, matrix,{-1}]; (*due to @KGLR*)
mat2 = Abs[Inverse[Drop[mat1, 1, 1]]] // N;
sa = SparseArray[select[mat2, .1, .6]];
weightedG = Graph[sa["NonzeroPositions"],  EdgeWeight -> sa["NonzeroValues"],DirectedEdges -> True];
GraphPlot[weightedG]

Even if I use VertexLabels->"Name" in Graph[...], the names (vertex numbers do not appear) on the graph. Most important of all, I like to use the column names a1, c5, d1,... as vertex names.

1. How can I make the entire process described above an automatic procedure? That is to say: when I plot the directed graph after row/column elimination, I like to keep track of the vertex string names and see those names on the final graph.

2. A more difficult task is: assigning the string column names to group names such as G1={a1,c5,d1}, G2={d2,d3}, G3={d4,d5} and zoom in the part of the graph including only selected groups such as {G1, G3}, meaning that group 2 G2 is excluded from the final digraph.

Answer 1

select[matrix_, lB_, uB_] := matrix  Map[Boole[lB <= # <= uB] &, matrix, {-1}];

mat1 = {{"", a1, c5, d1, d2, d3, d4, d5}, 
   {a1, 2, 0, 5, 3, 1, 5, 5}, 
   {c5, 1, 3, 1, 5, 0, 5, 4},
   {d1, 2, 3, 3, 5, 1, 4, 5},
   {d2, 2, 0, 5, 1, 5, 0, 4}, 
   {d3, 5, 0, 5, 5, 1, 2, 4}, 
   {d4, 2, 0, 4, 0, 1, 0, 4},
   {d5, 3, 2, 0, 4, 0, 1, 5}};

vnames = Rest[mat1[[1]]]

{a1, c5, d1, d2, d3, d4, d5}

1. You can construct a WeightedAdjacencyMatrix from mat2 replacing 0.s with ∞:

mat2 = Abs[Inverse[Drop[mat1, 1, 1]]] // N;

wam = select[mat2, .1, .6] /. 0. -> ∞;

MatrixForm @ wam

Use wam with WeightedAdjacencyGraph with vnames as the first argument:

wag = WeightedAdjacencyGraph[vnames, wam,
  VertexLabels -> "Name", EdgeLabels -> "EdgeWeight"]

2. "zoom in the part of the graph including only selected groups":

{G1, G2, G3} = {{a1, c5, d1}, {d2, d3}, {d4, d5}};

HighlightGraph[wag, Subgraph[wag, #] & /@ {G1, G2, G3}]

HighlightGraph[wag, Subgraph[wag, Join[G1, G3]], GraphHighlightStyle -> "Thick"]

Use VertexDelete to delete the vertices in G2 from wag:

vcoords = AssociationThread[vnames, GraphEmbedding[wag]];

Graph[VertexDelete[wag, G2], VertexCoordinates -> {v_ :> vcoords @ v}]