Visualizing weighted directed graphs

I will use IGraph/M for the following answer.

Here's a complete directed graph on 100 with weighted edges. There are lots of edges with small weights and a few with large weights.

graph = CompleteGraph[100, DirectedEdges -> True, 
   EdgeWeight -> RandomVariate[ExponentialDistribution[5], 100 99]];

We will:

• remove edges below a certain weight threshold
• style the remaining edges based on weight by adjusting their opacity and thickness

First, I need to utility functions.

Threshold an array (replace values below the threshold by zero). Threshold does not seem to work on sparse arrays.

threshold[arr_, th_] := arr UnitStep[arr - th]

Scale the elements of an array so that the largest is 1.

scale[arr_] := arr/Max@Abs[arr]

Now we convert the graph to a weighted adjacency matrix, threshold, convert back, then style it.

IGWeightedAdjacencyGraph[
   VertexList[graph], 
   threshold[WeightedAdjacencyMatrix[graph], 0.9], 
   GraphLayout -> "CircularEmbedding", ImageSize -> Large
] //
IGEdgeMap[
  Directive[AbsoluteThickness[5 #], Opacity[#], Arrowheads[0.05 #]] &,
  EdgeStyle -> scale@*IGEdgeProp[EdgeWeight]
]

We could use a smaller threshold to include more edges, but compute the opacity based on a power of the weight, to emphasize strong edges and fade out weak ones.

IGWeightedAdjacencyGraph[VertexList[graph], 
  threshold[WeightedAdjacencyMatrix[graph], 0.7], 
  GraphLayout -> "CircularEmbedding", ImageSize -> Large] //     
 IGEdgeMap[
  Directive[AbsoluteThickness[5 #], Opacity[#^3], 
    Arrowheads[0.05 #]] &, EdgeStyle -> scale@*IGEdgeProp[EdgeWeight]]

These choices about visualization are yours to make.