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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?

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]]}]