[assumes Code-01]

k=2, r=1

In[]:=
CANearestNeighborGraph[2,1,Range[0,2^(2^3)-1,2],​​EdgeStyle->Gray,​​VertexStyle->Directive[EdgeForm[Opacity[.5]],LightGray]]​​IsomorphicGraphQ[%,GridGraph[Table[2,7]]]
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True
[See BusyBeaver-01]
In[]:=
Table[If[MissingQ[#[[2]]],Nothing,#]&[n->FirstPosition[Total/@CellularAutomaton[n,{{1,1,1,1,1,1,1},0},{20,All}],0]],{n,0,255,2}]
Out[]=
{0{2},4{2},8{3},32{2},36{2},40{3},64{3},72{3},96{3},104{3},128{5},136{8},160{5},168{8},192{8},224{8}}
In[]:=
ArrayPlot[CellularAutomaton[#,{{1,1,1,1,1,1,1},0},{10,All}],Mesh->True,ImageSize->Tiny]&/@Keys[%]
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
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In[]:=
With[{posweights=Select[Association[Thread[#->Map[​​Function[v,quickLifetime[{1,1,1,1,1,1,1},20][v]​​],#]]]&[Range[0,2^(2^3)-1,2]],#>0&]},CANearestNeighborGraph[2,1,Keys[posweights],​​posweights,EdgeStyle->Gray,​​VertexStyle->Directive[EdgeForm[Opacity[.5]],LightGray],​​GraphLayout->{"LayeredDigraphEmbedding","RootVertex"->0},​​VertexSize->KeyValueMap[#1->(1/4)Sqrt[#2]&,posweights],​​VertexLabels->Placed[Automatic,Center],​​PerformanceGoal->"Quality"​​]​​]
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With[{posweights=Select[Association[Thread[#->Map[​​Function[v,quickLifetime[{1},20][v]​​],#]]]&[Range[0,2^(2^3)-1,2]],#>0&]},CANearestNeighborGraph[2,1,Keys[posweights],​​posweights,EdgeStyle->Gray,​​VertexStyle->Directive[EdgeForm[Opacity[.5]],LightGray],​​GraphLayout->{"LayeredDigraphEmbedding","RootVertex"->0},​​VertexSize->KeyValueMap[#1->(1/4)Sqrt[#2]&,posweights],​​VertexLabels->Placed[Automatic,Center],​​PerformanceGoal->"Quality"​​]​​]
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In[]:=
With[{posweights=Select[Association[Thread[#->Map[​​Function[v,quickLifetime[{1,0,1,1},20][v]​​],#]]]&[Range[0,2^(2^3)-1,2]],#>0&]},CANearestNeighborGraph[2,1,Keys[posweights],​​posweights,EdgeStyle->Gray,​​VertexStyle->Directive[EdgeForm[Opacity[.5]],LightGray],​​GraphLayout->{"LayeredDigraphEmbedding","RootVertex"->0},​​VertexSize->KeyValueMap[#1->(1/4)Sqrt[#2]&,posweights],​​VertexLabels->Placed[Automatic,Center],​​PerformanceGoal->"Quality"​​]​​]
Out[]=
In[]:=
Table[If[MissingQ[#[[2]]],Nothing,#]&[n->FirstPosition[Total/@CellularAutomaton[n,{{1,0,1,1},0},{20,All}],0]],{n,0,255,2}]
Out[]=
{0{2},8{3},32{3},36{3},40{4},64{3},96{4},104{5},128{2},136{3},160{3},164{3},168{4},192{3},224{4}}
In[]:=
ArrayPlot[CellularAutomaton[#,{{1,0,1,1},0},{5,All}],Mesh->True,ImageSize->Tiny]&/@Keys[%32]
Out[]=

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In[]:=
With[{posweights=Select[Association[Thread[#->Map[​​Function[v,quickLifetime[{1,1,1,1,1},20][v]​​],#]]]&[Range[0,2^(2^3)-1,2]],#>0&]},CANearestNeighborGraph[2,1,Keys[posweights],​​posweights,EdgeStyle->Gray,​​VertexStyle->Directive[EdgeForm[Opacity[.5]],LightGray],​​GraphLayout->{"LayeredDigraphEmbedding","RootVertex"->0},​​VertexSize->KeyValueMap[#1->(1/4)Sqrt[#2]&,posweights],​​VertexLabels->Placed[Automatic,Center],​​PerformanceGoal->"Quality"​​]​​]

k=2, r=3/2

[[ What is the probability to get to the maximum fitness by random choices of a single walker? ]]
This has speciated....
These are the “uncommitted” cases: (all variants of universal common ancestor)
I.e. you can reach everything from 0, assuming you allow all mutations.....

Lifetime distribution

{1,1,1,0,1,1} case:

Can allow a different level of non-optimality at each step

Assume that you always pick the “top fitness” choices at each step
Keep top 2 fitnesses at each step:
Can get rid of “lateral edges”....
<<<< We want to take the original graph, and simply prune things one can’t reach by maximum-fitness-at-each-step moves >>>>

k=2, r=1, multiparent