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I always wonder how many inbuilt functions does Mathematica have (of course you can google for it) and how they are connected with each other! So I tried this (v10.1).

SetDirectory[$InstallationDirectory<>"/Documentation/English/System/ReferencePages/Symbols"]
comms = FileNames[];
ncomms = Length[comms]

4613

I believe this is the total number of documented functions. Now the second part where I am stuck in. How to find the connections among them?

What I am thinking is to get the list of functions from the See Also section of an example. Here I do one manually. Let's say I start with Plot. See Also in Plot.nb contains ({DiscretePlot, ListLinePlot, ParametricPlot, PolarPlot, Plot3D, ContourPlot, Graphics, Show}). So I store them.

link["Plot"] = {"DiscretePlot", "ListLinePlot", "ParametricPlot" , 
                "PolarPlot", "Plot3D", "ContourPlot", "Graphics", "Show"};

Now I want to scan each element of the list. For example, I take Plot3D and Graphics

link["Plot3D"] = {"ListPlot3D", "ContourPlot", "DensityPlot", "ParametricPlot3D",
                  "Graphics3D", "ListSurfacePlot3D", "Plot", "Show"};

link["Graphics"] = {"Plot", "ListPlot", "ListLinePlot", "ParametricPlot", "DensityPlot",
                    "ArrayPlot", "RegionPlot", "ContourPlot", "Show", "Graphics3D",
                    "Image", "Import", "Sound"};

Now combine and map them

map = Join[DirectedEdge[num["Plot"], num[#]] & /@ link["Plot"],
           DirectedEdge[num["Plot3D"], num[#]] & /@ link["Plot3D"],
           DirectedEdge[num["Graphics"], num[#]] & /@ link["Graphics"]];

Graph[map, VertexLabels -> Table[n -> sets[[n]], {n, nsets}], ImagePadding -> 10]

enter image description here

How can I put this whole thing in a Mathematica code?

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  • 3
    $\begingroup$ You can start with: WolframLanguageData["Plot", "RelationshipCommunityGraph"] $\endgroup$ – Kuba Aug 10 '16 at 11:56
  • $\begingroup$ closely related: How to get complete Documentation Center graph of guide pages? $\endgroup$ – Kuba Aug 10 '16 at 12:08
  • $\begingroup$ thanks @Kuba, didn't know about WolframLanguageData. I like Leonid's answer because it will work for the older version as well. $\endgroup$ – Sumit Aug 10 '16 at 12:55
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(* docs on my system (10.3, Windows) *)
base = FileNameJoin[{$InstallationDirectory, "Documentation",
    "English", "System", "ReferencePages", "Symbols"}];

FileNames[FileNameJoin[{base, "*"}]] // Length

4811

(* symbols in See Also section *)
also[name_String] :=
 Module[{import},
  import = Import[
    FileNameJoin[{base, name <> ".nb"}], {"Cells", "SeeAlso"}];
  Cases[import, TextData[s_String] :> s, Infinity]]

also["I"]

{"Complex", "Re", "Im", "ComplexExpand", "GaussianIntegers"}

edges[start_String, n_Integer] :=
 Reap[Module[{visited = {}},
     Nest[
      Function[visiting,
       Module[{temp},
        temp = Join @@ Table[
           Sow[DirectedEdge[v, #] & /@ also[v]], {v, visiting}];
        visited = Join[visited, visiting];
        Complement[Last /@ temp, visited]]],
      {start}, n]]][[2, 1]] // Flatten

graph[edges : {__DirectedEdge}, start_String, opts___] :=
 Graph[edges, opts,
  VertexLabels -> "Name",
  VertexStyle -> {start -> Red}]

Example 1

With[{x = "I"}, graph[edges[x, 2], x]]

enter image description here

Example 2

(* this took me 40 seconds *)
e5 = edges["Plot", 5];

g = graph[e5, "Plot",
  VertexLabels -> None,
  EdgeStyle -> Directive[Opacity[.2]],
  EdgeShapeFunction -> ({Arrowheads[.01], Arrow[#]} &)];

Through[{VertexCount, EdgeCount}@g]

{1463, 4192}

Majority of vertices are not reachable from a general source (dark red area). Note that: a) graph is directed, and b) crawler stopped after five steps.

dm = GraphDistanceMatrix[g];
dm // MatrixPlot

enter image description here

max = Max[dm /. Infinity -> 0]

13

pos = Position[dm, max];
pairs = VertexList[g][[#]] & /@ pos;

Example of a longest shortest path (via pairs). Starting in Help with Repeated, 13 clicks are needed at least to get to Arrow.

sp = FindShortestPath[g, "Repeated", "Arrow"];
(* {"Repeated", "BlankSequence", ... "Arrow"} *)

HighlightGraph[g, PathGraph[sp, DirectedEdges -> True]]

enter image description here

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  • $\begingroup$ This method is awesome! In lots of aspects it's better than mine! would you mind me to edit my answer a bit to include some awesome part of your answer? (like Import stuff) $\endgroup$ – Wjx Aug 10 '16 at 15:31
  • $\begingroup$ Also, this method has one small drawback in the crawler part: you actually can store something to aviod repetitive calculation and import. This may speed things up a lot I suppose? :) $\endgroup$ – Wjx Aug 10 '16 at 15:36
  • $\begingroup$ @Wjx No, I wouldn't mind, and yes, I've been thinking about the crawler ... will update. $\endgroup$ – BoLe Aug 10 '16 at 17:35
  • $\begingroup$ However, result for cosine is nothing like WolframLanguageData example. $\endgroup$ – BoLe Aug 10 '16 at 18:00
  • $\begingroup$ I've consolidated the code and added an example. $\endgroup$ – BoLe Aug 11 '16 at 8:33
8
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There is a way for you to find out the relations without internet, and I think it fit your need more as they are actually extracted directly from those "See Also" links.

I'm currently using v11, 5000+ Functions, it takes about 7 min before it finishes the main evaluation, and another 10 min or so for plotting the community graph out, but it's totally internet free, thus can be a bit more helpful when under bad net conditions.

dir = FileNameJoin[{$InstallationDirectory, 
    "Documentation\\English\\System\\ReferencePages\\Symbols"}];
files = Import[dir];

rules = Flatten[Function[{name}, Thread[StringSplit[name, "."][[1]] -> 
   Block[{f = Import[FileNameJoin[{dir, name}]], p}, 
    p = Position[f, "See Also"]; 
     If[p != {}, Cases[#, TextData[s_String]:>s] &@
      Extract[f, Drop[p[[-1]], -5]], {}]]]] /@ files];

Graph@rules
CommunityGraphPlot@%

This method follows how we humans know about the links between functions-----via checking the "See Also" part of the documentation of course. So we can extract those part out one by one, make rules out of them, then use Graph and CommunityGraphPlot to visualize them.

The result is stylish(If we view in the Abstract Painting's style :P) and the structure is clear(Though a bit messy): Mathematica's core functions are closely related!!!

graph

community graph

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  • $\begingroup$ Thanks @Wjx, I am indeed looking for a method which does not involve the internet. I added that to the title. $\endgroup$ – Sumit Aug 10 '16 at 13:55
  • $\begingroup$ So will this be what you need? @Sumit :) $\endgroup$ – Wjx Aug 10 '16 at 14:10
  • $\begingroup$ yup. I will accept it few hours later - just in case someone has some other thoughts ;) $\endgroup$ – Sumit Aug 10 '16 at 14:57
  • $\begingroup$ I was wondering whether the complete relationship was connected, as in a single component graph ... Did you identify some of the symbols in other components? $\endgroup$ – BoLe Aug 10 '16 at 15:00
  • $\begingroup$ ImageIdentify thinks that these graphs are a virus and a bobby pin, respectively. $\endgroup$ – Karsten 7. Aug 10 '16 at 18:12
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Something like this:

g = SimpleGraph@Graph[
    Catenate[
     Thread /@ 
      Normal@DeleteMissing@
        WolframLanguageData[WolframLanguageData[], "RelatedSymbols", "EntityAssociation"]
     ],
    DirectedEdges -> False
    ];

It's slow, like most new *Data functions, but it works.

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  • 1
    $\begingroup$ You can get a bunch of smaller graphs that show some connections with WolframLanguageData[All, "RelationshipGraph"] -- so not exactly what the OP asked for but interesting visualisations anyway. It doesn't have all the functions (WLD[f, "RelationshipGraph"] doesn't show f in its graph). Makes for a nice puzzle, as well... $\endgroup$ – Gerli Aug 11 '16 at 8:56

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