Exporting organization of Mathematica's documentation to a JSON file

Question

In this answer, @Vitaliy Kaurov explores and utilizes new Mathematica's functionality WolframLanguageData:

graphicssym = 
  EntityList[EntityClass[
    "WolframLanguageSymbol", {"FunctionalityArea", "GraphicsFunctions"}]];

linktrails = 
  EntityValue[graphicssym, EntityProperty["WolframLanguageSymbol", "LinkTrails"]];

edges = 
  Flatten[(DirectedEdge @@@ Partition[#, 2, 1]) & /@ 
     Select[Flatten[linktrails, 1], Length[#] > 1 &]];

verts = Flatten[List @@@ edges] // Union;

Graph[edges, 
  VertexLabels -> Placed["Name", Tooltip], 
  VertexStyle -> ((# -> If[StringMatchQ[#, "*ref/*"], Red, Green]) & /@ verts)]

As you can see, there two types of nodes: "green" for guides, and "red" for reference pages. This covers of course only a small subset of Mathematica documentation, since "FunctionalityArea" is limited to "GraphicsFunctions". But this is convenient for visual exploration, certainly much better than a graph with hundreds of thousands nodes, so I like this as a test example.

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I would like to do some visualization of Mathematica's documentation structure with external tools. In order to do this, I want to obtain information presented in the graph above inside a JSON file with the following format:

{
  "groups":[
    {"name":"guide"},
    {"name":"ref"}
  ],
  "nodes":[
    {"name":"Finance","group":0},
    {"name":"StringManipulation","group":0},
    {"name":"Raster","group":1},
    {"name":"CapForm","group":1},
    ...
  ],
  "links":[
    {"source":1,"target":0},
    {"source":2,"target":0},
    {"source":3,"target":0},
    {"source":3,"target":2},
    ...
  ]
}

• "groups" are simply two types of pages of Mathematica documentation, "guide", and "ref";
• "nodes" is a list of nodes that contain the node name and index of the node's group;
• "links" is a list of links between nodes, and nodes are determined by their indexes in their list.

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I tried to incorporate @Szabolcs 's code from another answer into my solution:

g = RandomGraph[BarabasiAlbertGraphDistribution[100, 1]]

names = VertexList[g];
groups = VertexDegree[g]; (* let's try degree-based colouring *)

Export[
  "graph.json",
  {
    "nodes" -> MapThread[{"name" -> #1, "group" -> #2} &, {names, groups}],
    "links" -> 
       ({"source" -> #1 - 1, "target" -> #2 - 1, "value" -> 1} &) @@@ EdgeList[g]
  },
  "JSON"]

---

However, I couldn't get all bits and pieces working. Hope some of you could help.