Graphs generated by Dynamical Systems

NormalizeGraph[ss_]:=Module[{EE,E2,VV,vl,nn,rules,V1=VertexList[ss],E1=EdgeRules[ss]},E2={};Do[If[Not[E1[[k,1]]==E1[[k,2]]],E2=Append[E2,E1[[k]]]],{k,Length[E1]}];E1=E2;
vl=VertexList[ss];nn=Length[vl];rules=Table[vl[[k]]->k,{k,nn}];EE=E1/.rules;VV=V1/.rules;
UndirectedGraph[Graph[VV,EE]]];
NaturalGraph[nn_,T_]:=Module[{},UndirectedGraph[Graph[Union[Flatten[Table[y=T[[k]][x];
If[x==y,{},x->y],{x,0,nn-1},{k,Length[T]}]]]]]];
NaturalGraphPlot[s_]:=Module[{ss,u,Q,S1,S2,Pn,Qn,VD,mVD,mED,ED,EE},
ss=NormalizeGraph[s];
VD=VertexDegree[ss];mVD=Max[VD];VD=VD/mVD;
      EE=EdgeList[ss];ED=Table[VD[[EE[[k,1]]]]+VD[[EE[[k,2]]]],{k,Length[EE]}];
mED=Max[ED]; ED=ED/mED;
line[{a_,b_}]:=Line[{a,b}];
P=GraphPlot[ss][[1,1,1]];Pn=Length[P];
u=Normal[GraphPlot[ss][[1,1]]][[1,2]];
Q=Table[u[[k,1]],{k,Length[u]}];Qn=Length[Q];
S1=Table[{Yellow,PointSize[VD[[k]]*0.02],Point[P[[k]]]},{k,Pn}];
S2=Table[{Red,Thickness[ED[[k]]*0.005],line[Q[[k]]]},{k,Qn}];
Show[Graphics[{S2,S1}],ImageSize->400,
Background->Black,Axes->False,AspectRatio->1]];

Manipulate[Module[{u,v,T1,T2},Which[
u=Floor[1+100*(α-0.9)];v=Floor[1+100*(β-0.9)];
r==4,T1[x_]:=Mod[u x+a,n];T2[x_]:=Mod[v x+b,n],
r==3,T1[x_]:=Mod[Floor[x^α+a],n];T2[x_]:=Mod[Floor[x^β+b],n],
r==7,T1[x_]:=Mod[Floor[x^α+x^β +b*x+a],n];T2[x_]:=Mod[x,n],
r==1,T1[x_]:=Mod[Floor[x^u+a],n]; T2[x_]:= Mod[Floor[x^v+b],n],
r==5,T1[x_]:=Mod[Floor[x^u+x^v+b*x+a],n];T2[x_]:=Floor[x],
r==6,T1[x_]:=Mod[Floor[u^x+v^x+b*x+a],n];T2[x_]:=Floor[x],
r==2,T1[x_]:=Mod[ Floor[2^x+a*x+b],n]; T2[x_]:=Mod[u x+v,n]];
Quiet[NaturalGraphPlot[NaturalGraph[n,{T1,T2}]]]],
{{n,10},10,300,1},{{α,1},0.9,1.1},{{β,1},0.9,1.1},{{a,1},1,30,1},{{b,2},1,30,1},
{{r,3,"Graph:"},{5->"poly 1",1->"poly 2",6->"exp 1",2->"exp 2",3->"alpha",7->"alpha 1",4->"lin"},ControlType->SetterBar}]

Graphs generated by finitely many transformations on a finite ring have been introduced by

Ghachem and Knill.

References:

[1] M. Ghachem and O. Knill, "Simple Rules"  http://arxiv.org/abs/1009.2292

[2]M. Ghachem and O. Knill,  "Deterministic Watts-Strogatz"  http://arxiv.org/abs/1111.5395

Contributed by: Oliver Knill, Harvard University, knill@math.harvard.edu