How do I initialise a compound expression in Animate? Or, how do I have an expression evaluate on startup?

I have written a notebook with a few functions I defined.
These functions describe a way of generating graphs using 2 inputs.
e.g. graph[n_,s_] := Graph[...]

I have also defined a function animating the process of drawing the graph.

animateGraph[n_Integer, s_List, p_Integer : 0] :=
 Module[{
   pathList = path[aiList[n, s, p]],
   verts = Range[n] - 1,
   coords = vertCoords[n]},
  Animate[
   Graph[
    verts,
    pathList[[;; i]],
    VertexLabels -> Automatic,
    VertexCoordinates -> coords,
    EdgeShapeFunction -> esf],
   {i, 0, Length@pathList, 1},
   AnimationRunning -> False,
   AnimationRepetitions -> 1]]  

It just takes more and more of the vertices until the graph is complete.
The problem, however, is that when I close the kernel and start again, the Animator is still trying to use the temporary variables from the last time it was evaluated. For example, with animateGraph[5,{1}], the Animator window displays:

Graph[{0, 1, 2, 3, 4, 5}, pathList$12323[[1 ;; 5]], 
 VertexLabels -> Automatic, VertexCoordinates -> coords$12323, 
 EdgeShapeFunction -> esf]  

instead of the animation I want.
And I get a repeated error message: Symbol::argx
I want to upload my work to the Community blog, but the notebook doesn’t display the animations correctly. Is there a way I can have each call of animateGraph evaluate on startup so that new temporary variables are created?
I’m not sure if this helps, but I did try evaluating the definitions of the relevant functions in the Initialization option of Animate, but that doesn’t seem to work either.

For completeness, I will include all the definitions needed to run graph, and animateGraph. If you want to generate a random graph, or animateGraph, just evaluate graph[ranomnS[]], or animateGraph[randomnS[]]

j[i_Integer, k_Integer] := Mod[i, k, 1];
sj[i_Integer, s_List] := s[[j[i, Length@s]]];

ai[0, n_Integer, s_List, p_Integer : 0] := p~Mod~n;
ai[i_Integer, n_Integer, s_List, 
   p_Integer : 0] := (ai[i - 1, n, s, p] + sj[i, s])~Mod~n;

t[n_Integer, s_List] := (n*Length[s])/GCD[n, Total[s]];

aiList[n_Integer, s_List, p_Integer : 0] :=
  Table[ai[i, n, s, p], {i, 0, t[n, s]}];

randomnS[nMax_Integer : 20, kMax_Integer : 20] :=
  Module[
   {n = RandomInteger[{3, nMax}],
    k = RandomInteger[{1, kMax}],
    s},
   s = RandomInteger[{-n, n}, k];
   Unevaluated[Sequence[n, s]]];

path[vertices_List, directed_Symbol : Rule] := 
  directed @@@ Partition[vertices, 2, 1];

vertCoords[n_] := 
  Table[{Cos[(2 [Pi] i)/n], Sin[(2 [Pi] i)/n]}, {i, n}];

esf[el_, ___] := {Black, Arrowheads[0.02], Arrow[el, 0.05]};  
Options[graph] = {"Union" -> False};
graph[n_Integer, s_List, p_Integer : 0, OptionsPattern[]] :=
  Graph[
   Range[n] - 1,
   If[OptionValue["Union"], Union, (# &)]@path[aiList[n, s, p]],
   VertexLabels -> Automatic,
   VertexCoordinates -> vertCoords[n],
   EdgeShapeFunction -> esf,
   ImageSize -> Medium];  

Please let me know if there is more information I should add.

Manipulate[
 animateGraph[n_Integer, s_List, p_Integer : 0] :=
  
  DynamicModule[{i = 1, pathList = path[aiList[n, s, p]], 
    verts = Range[n] - 1, coords = vertCoords[n]},
   Dynamic[
    Refresh[Graph[verts, 
      pathList[[;; If[i < Length[pathList], Pause[x]; i = i + 1, i]]],
       VertexLabels -> Automatic, VertexCoordinates -> coords, 
      EdgeShapeFunction -> esf]]
    ]];
 animateGraph[n, s],
 Control[{{n, 1, "Points"}, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 
    13, 14, 15, 16, 17, 18, 19, 20}}]
 , Control[{{s, {1}, "List"}, ControlType -> InputField}], 
 Control[{{x, 0.1, "Delay"}, 0, 1}], 
 Initialization -> (j[i_Integer, k_Integer] := Mod[i, k, 1];
   sj[i_Integer, s_List] := s[[j[i, Length@s]]];
   
   ai[0, n_Integer, s_List, p_Integer : 0] := p~Mod~n;
   ai[i_Integer, n_Integer, s_List, 
     p_Integer : 0] := (ai[i - 1, n, s, p] + sj[i, s])~Mod~n;
   t[n_Integer, s_List] := (n*Length[s])/GCD[n, Total[s]];
   
   aiList[n_Integer, s_List, p_Integer : 0] := 
    Table[ai[i, n, s, p], {i, 0, t[n, s]}];
   
   path[vertices_List, directed_Symbol : Rule] := 
    directed @@@ Partition[vertices, 2, 1];
   
   vertCoords[n_] := 
    Table[{Cos[(2 [Pi] i)/n], Sin[(2 [Pi] i)/n]}, {i, n}];
   
   esf[el_, ___] := {Black, Arrowheads[0.02], Arrow[el, 0.05]};
   Options[graph] = {"Union" -> False};
   )]