How to plot a signal (function) on a graph (object of graph theory)

Question

Given a graph of n vertices, it is possible to plot a discrete signal (or function) of n samples on the vertices of the graph, so that one can visualize the features of the signal on the graph (see the attached image)?

Answer 1

g = PetersenGraph[];

(* get the 3d coordinates of the graph vertices *)
coords = Append[#, 0] & /@ GraphEmbedding[g];
points = Point[coords];

(* create lines between the edges *)
connections = EdgeList[g];
lines = Line[coords[[#]]] & /@ (connections /. UndirectedEdge -> List);

(* generate some signals on each vertex as lines pointing upwards *)
SeedRandom[1];
signals = RandomReal[1, VertexCount@g];
signalsLines = MapThread[
  Line[{#1, Append[Most@#1, #2]}] &,
  {coords, signals}];

(* draw it all in ortho projection *)
Graphics3D[{
  {Red, Dashed, lines},
  {Red, PointSize[Large], points},
  {Blue, signalsLines}
  }, Boxed -> False, ViewProjection -> "Orthographic"]

Answer 2

SeedRandom[1]
signals = RandomReal[1, 10];

To get a 3D graph that looks like the one in OP, we can use signals to specify a custom VertexShapeFunction and use it with PetersenGraph:

PetersenGraph[
 EdgeStyle -> Directive[Dashed, Red],
 VertexCoordinates -> Append[0] /@ GraphEmbedding[PetersenGraph[]],
 EdgeShapeFunction -> "Line",
 VertexShapeFunction -> ({Thick, Blue, 
     Line[{#, # + {0, 0, signals[[#2]]}}], Red, Sphere[#, .03]} &),
 ImageSize -> 500
 ]

More generally, we can use signals as the setting for the options VertexSize and/or VertexStyle

g0 = PetersenGraph[ImageSize -> 300];

g1 = PetersenGraph[ImageSize -> 300,
   VertexSize -> {v_ :> signals[[v]]},
   VertexStyle -> {v_ :> ColorData["Rainbow"]@Rescale[signals][[v]]},
   EdgeStyle -> Directive[Dashed, Red],
   EdgeShapeFunction -> "Line",
   BaseStyle -> FaceForm[Opacity[.5]]];

Row[{g0, g1}]

To get a 3D graph object simply wrap g1 with Graph3D. Use a custom VertexShapeFunction if you need to depict vertices as lines:

Row[{Graph3D @ g1,
  Graph3D[g1, 
    VertexShapeFunction -> ({Thick, Line @ {#, # + {0, 0, signals[[#2]]}}} &)]}]

We can also use a more flexible custom VertexShapeFunction which allows any 3D graphics primitive as vertex shape:

ClearAll[vShapeF, tF]

tF[x_] := Translate[
    Scale[x, If[FreeQ[_Sphere]@x, {.1, .1, #3[[3]]}, #3], {0, 0, 0}], #] &;

vShapeF[prim_: Automatic, d___] := 
 Module[{pnt = {AbsolutePointSize[5], Black, Opacity[1], Point[{0, 0, 0}]}, 
   dir = Sequence[Opacity[.5], AbsoluteThickness[2], CapForm[None], d]},
Switch[prim,
  Automatic | "Automatic" | Sphere | "Sphere", 
  tF[{dir, Sphere[], pnt}],
  Line | "Line", tF[{dir, Line[{{0, 0, 0}, {0, 0, 1}}], pnt}],
  Tube | "Tube", tF[{dir, Tube[{{0, 0, 0}, {0, 0, 1}}, 1], pnt}],
  Cylinder | "Cylinder", 
  tF[{dir, Cylinder[{{0, 0, 0}, {0, 0, 1}}, 1], pnt}],
  Cuboid | "Cuboid", tF[{dir, Cuboid[{-1, -1, 0}, {1, 1, 1}], pnt}],
  _, tF[{dir, prim,  pnt}]]]

Examples:

triangleWaveCube = ChartElementData["TriangleWaveCube", 
   "AngularFrequency" -> 5, "RadialAmplitude" -> 0.4][{{-1, 1}, {-1, 1}, {0, 1}}];

Multicolumn[
 Graph3D[g1,  VertexShapeFunction -> vShapeF[ToExpression @ #], 
    BoxRatios -> If[# == "Automatic", Automatic, {1, 1, 1/2}], 
    PlotLabel -> Style[#, 16, Black], 
    ImageSize -> 400] & /@ 
  {"Automatic", "Line", "Tube", "Cylinder", "Cuboid", "triangleWaveCube"}, 
  2, Appearance -> "Horizontal"]