How to link a variable to vertices of a directed graph using bubles with different sizes

Question

This Code produces a directed graph:

SeedRandom[12];
mat = RandomReal[{0, 2}, {5, 5}];
select[matrix_, lB_, uB_] := matrix*Map[Boole[lB <= # <= uB] &, matrix, {-1}];
sa = SparseArray[select[mat, .1, .4]];
weightedG = Graph[sa["NonzeroPositions"], EdgeWeight -> sa["NonzeroValues"], 
   DirectedEdges -> True, VertexLabels -> Automatic]

I like to attach a variable:

emp = {10, 45, 2, 1, 49};

to each vertex using bubbles with different sizes based on the variable emp, that is, the bubble size of vertex 1 should be a normalized size (i.e., 10/107), for vertex 2 (45/107), and so on. Elements of the variable emp are associated with five vertices {1,2,3,4,5}, respectively.

The final directed graph should be one with vertices of different bubble sizes.

Answer 1

You can use emp as the setting for VertexSize:

Graph[sa["NonzeroPositions"], 
 VertexSize -> {v_ :> ({#, #} & @ Normalize[emp, Total][[v]])},
 PerformanceGoal -> "Quality", (* so that arrow heads are not covered by vertex disks *) 
 ImagePadding -> 10, (* so that vertex labels are not clipped *)
 EdgeWeight -> sa["NonzeroValues"], 
 DirectedEdges -> True,
 VertexLabels -> Automatic]

Answer 2

With your definitions:

emp = {10, 45, 2, 1, 49};
normemp = Thread[Range[Length[#]] -> Normalize[#, Total]] &@emp;

ClearAll[f]
f[{x_, y_}, v_, {w_, h_}] := Disk[{x, y}, v /. normemp]

Graph[
 sa["NonzeroPositions"], EdgeWeight -> sa["NonzeroValues"], 
 DirectedEdges -> True, VertexLabels -> Automatic, 
 VertexShapeFunction -> f
]