Convert GraphPlot[]s with many nodes into something that's human-understandable

Question

This is more of an abstract/creative problem. I think everyone who has played around with GraphPlot, ended up with an image that looks like this, at least once in their life:

I am looking for general solutions to convert many connections -> many connections into something that's human understandable.

Here's my code above:

d = DictionaryLookup[RegularExpression["[a-z]{1,5}"]];
d = Take[d, 1000];
g = {};
Do[Do[If[Abs[StringLength[d[[i]]] - StringLength[d[[j]]]] > 1, 
   Continue[]];
  If[EditDistance[d[[i]], d[[j]]] == 1, 
   AppendTo[g, d[[i]] -> d[[j]]];
   AppendTo[g, d[[j]] -> d[[i]]]], {j, i + 1, Length[d]}], {i, 1, 
  Length[d]}]

Needs["GraphUtilities`"]
s = StrongComponents[g];
sc = Last[SortBy[s, Length]];
gc = Select[g, MemberQ[sc, #[[1]]] &];
GraphPlot[gc, 
 EdgeRenderingFunction -> ({Orange, Arrowheads[Small], 
     Arrow[#1, .07]} &)]

I would love something like this, where strongly connected nodes are grouped into single pie-chart vertices which represent the # of node connections over total connections.

Answer 1

Your graph has many vertexes, while the exemplary graph you link to does not and this is why it looks so nice. Styling with pie charts is not a problem at all:

g = RandomGraph[{15, 43}];

vfc[{xc_, yc_}, name_, {w_, h_}] := 
 Inset[PieChart[{VertexDegree[g, name], VertexList[g] // Length}, 
   SectorOrigin -> {Automatic, .7}, 
   ChartStyle -> "AvocadoColors"], {xc, yc}, Automatic, 
  VertexDegree[g, name]/30]

SetProperty[g, VertexShapeFunction -> vfc]

I by the way understand that you want more - to replace "heavy" sub-graphs by nodes reflecting their statistics. That could be done, but it is a bit messy and ambiguous, more of a research question. If I find time I may post a solution.

Here is another angle at a quicker analysis. Automatic spatial layouts Mathematica uses for graph carry a lot of information too. We can use FindClusters to analyse it. I will remove directed and multiple edges for clear picture. To see clustering of regions, after you execute your code do this:

n = 10;(*number of clusters*)
grr = Graph[Union[Sort /@ gc], DirectedEdges -> False, GraphStyle -> 
      "LargeNetwork", VertexSize -> 0];
cls = ListPlot[FindClusters[AbsoluteOptions[grr, VertexCoordinates][[2]], n],
      PlotStyle -> ColorData[3, "ColorList"]];
Show[grr, cls, BaseStyle -> PointSize[.01]]