How to color each edge of a graph with two colors?

Question

This is a chord visualization taken from here. The corresponding code for visualization is

g = ExampleData[{"NetworkGraph", "LesMiserables"}, "FullGraph"]
v = VertexList[g]
e = EdgeList[g];
r = 10;
tsep = 1.0;
ang = 2 Pi/Length[v] + 0.0;
gelt2 = Table[vind1 = Position[v, e[[i, 1]]][[1, 1]];
   vind2 = Position[v, e[[i, 2]]][[1, 1]];
   {Opacity[0.5], RGBColor[0.6, 0.729, 1], 
    BSplineCurve[{{(r - 0.5)*Cos[ang*vind1], (r - 0.5)*
        Sin[ang*vind1]}, {0, 
       0}, {(r - 0.5)*Cos[ang*vind2], (r - 0.5)*
        Sin[ang*vind2]}}]}, {i, 1, Length[e]}];
gdyn = Table[cv = v[[j]];
  tempe = EdgeList[g, cv \[UndirectedEdge] _];
  rot = (ang*j > Pi/2) && (ang*j < 3*Pi/2);
  Mouseover[
   (*if mouse not on top*)(*render the character name*)
   Rotate[Text[
     Style[(*Limit the character name to 8 characters only*)
      If[StringLength[cv] > 8, StringTake[cv, 8] <> ".", cv], 
      Medium], {(r + tsep)*Cos[ang*j], (r + tsep)*Sin[ang*j]}], 
    If[rot, ang*j - Pi, 
     ang*j]], {(*if mouse on top*)(*render the character name*)
    Rotate[
     Text[Style[cv, Medium, Blue, 
       Bold], {(r + tsep)*Cos[ang*j], (r + tsep)*Sin[ang*j]}], 
     If[rot, ang*j - Pi, ang*j]],(*render thick bsplines curves*)
    Table[vind1 = Position[v, tempe[[i, 1]]][[1, 1]];
     vind2 = Position[v, tempe[[i, 2]]][[1, 1]];
     {Thick, 
      BSplineCurve[{{(r - 0.5)*Cos[ang*vind1], (r - 0.5)*
          Sin[ang*vind1]}, {0, 
         0}, {(r - 0.5)*Cos[ang*vind2], (r - 0.5)*
          Sin[ang*vind2]}}]}, {i, 1, Length[tempe]}] (*end of thick b-
    spline table*)} (*end of Mouseover second argument*)
   ],(*end of Mouseover*){j, 1, Length[v]}];(*end of gdyn table*) 

The corresponding visualization is:

Now I wish to color each edge with two colors - the first half with one color and the second half with another color and all the edges from the same vertex should have the same color. A sample is shown below:

How can I do this?

Answer 1

Update 2: An alternative approach that gives better-looking curved edges:

ClearAll[eSF, vSF]
eSF[clr_Association] := (Quiet@GraphComputation`GraphPropertyChart[]; 
  GraphComputation`GraphChartDump`pEdge[blah, blah, blah, #1, #2]/.
   Style[circ_Circle, _] :> circ /.  Circle[aa_, bb_, cc_] :> 
    MapThread[Function[{x, y}, {x, Circle[aa, bb, y]}], 
     {clr /@ {First@#2, Last@#2}, Partition[Subdivide[## & @@ cc, 2], 2, 1]}]) &;

vSF[clr_Association] := Module[
  {off = If[-Pi/2 < ArcTan @@ # < Pi/2, Left, Right]},
  {clr @ #2, Text[Style[Framed[#2, FrameStyle -> None], 
       FontSize -> Scaled[.03]], #, {off, Center}, 
       ArcTan[#] (off /. {Left -> 1, Right -> -1})],
     PointSize[Large], Point@#}] &;

Example:

g = ExampleData[{"NetworkGraph", "LesMiserables"}, "FullGraph"];

vColors = AssociationThread[VertexList[g], 
   RandomSample[ColorData[{"Rainbow", {1, VertexCount@g}}] /@ 
     Range[VertexCount[g]]]];

SetProperty[g, {ImageSize -> Large, 
  GraphLayout -> "CircularEmbedding", 
  VertexShapeFunction -> vSF[vColors], 
  EdgeShapeFunction -> eSF[vColors]}]

Update: You can also use custom functions for the options EdgeShapeFunction and VertexShapeFunction:

ClearAll[eSf, vSf]
eSf[g_, cols_] := Module[{bsf = BSplineFunction[{#[[1]], 
     RegionNearest[Disk[Mean[#[[{1, -1}]]], Norm[#[[1]] - #[[-1]]]], {0, 0}], #[[-1]]}], 
     p1 = Subdivide[0, 1/2, 50], p2 = Subdivide[1/2, 1, 50]}, 
   {Thin, cols[[VertexIndex[g, #2[[1]]]]], Line[bsf /@ p1], 
     cols[[VertexIndex[g, #2[[2]]]]], Line[bsf /@ p2]}] &;
vSf[g_, cols_] := Module[{off = If[-Pi/2 < ArcTan @@ # < Pi/2, Left, Right]},
   {cols[[VertexIndex[g, #2]]], 
    Text[Style[Framed[#2, FrameStyle -> None], FontSize -> Scaled[.03]],
       #, {off, Center}, ArcTan[#] (off /. {Left -> 1, Right -> -1})],
    PointSize[Large], Point @ #}] &;

Example:

g = ExampleData[{"NetworkGraph", "LesMiserables"}, "FullGraph"];
cols = RandomSample[ColorData[{"Rainbow", {1, VertexCount@g}}] /@ Range[VertexCount[g]]]; 

SetProperty[g, {ImageSize -> Large, GraphLayout -> "CircularEmbedding", 
  VertexShapeFunction -> vSf[g, cols], EdgeShapeFunction -> eSf[g, cols]}]

You can add Epilog -> Circle[] in the second argument of SetProperty above to get:

Original answer:

You can use BSplineFunction:

cps1 = {{8, 5}, {0, 0}, {10, 1}};
Graphics[{Thick, Red, Line[BSplineFunction[cps1] /@ Subdivide[0, 1/2, 50]], 
   Blue, Line[BSplineFunction[cps1] /@ Subdivide[1/2, 1, 50]]}]