Select cases with a particular number of particles

In[]:=
ggg=Graph[Flatten[Table[{#,i}->{Last[ResourceFunction["BlockCellularAutomaton"][{{2,2}{1,1},{1,1}{2,2},{1,2}{1,2},{2,1}{2,1},{2,0}{0,2},{1,0}{1,0},{0,2}{2,0},{0,1}{0,1},{0,0}{0,0}},#1,1,i]],1-i},{i,0,1}]&/@Select[Tuples[{0,1,2},8],Total[Sign[#]]==5&]],VertexSize->{x_:>.2MaxBlob[First@x]}]
Out[]=
In[]:=
Length[WeaklyConnectedGraphComponents[ggg]]
Out[]=
224
In[]:=
Select[Tuples[{0,1,2},8],Total[Sign[#]]==5&]//Length
Out[]=
1792
In[]:=
1792/16
Out[]=
112

Want to remove components related to others by rotation...

In[]:=
Length/@GatherBy[{#,Sort[ResourceFunction["CanonicalListRotation"][First[#]]&/@VertexList[#]]}&/@WeaklyConnectedGraphComponents[ggg],Last]
Out[]=
{2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,2,2,1,2,1,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,1,2,2,1,2,2,2,2,2,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,1,1,2,2,2,2,2,2,2,1,2,2,2,2,2,2,1,2,2,2,2,2,2,1,1,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,1,2,1,2,2,1,2,1,1,1,1,2,1,1}
In[]:=
GraphUnion[#[[1,1]]&/@GatherBy[{#,Sort[ResourceFunction["CanonicalListRotation"][First[#]]&/@VertexList[#]]}&/@WeaklyConnectedGraphComponents[ggg],Last]]
GraphUnion
:A graph object is expected at position 1 in GraphUnion
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,114.
Out[]=
GraphUnion
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
In[]:=
GraphUnion@@(#[[1,1]]&/@GatherBy[{#,Sort[ResourceFunction["CanonicalListRotation"][First[#]]&/@VertexList[#]]}&/@WeaklyConnectedGraphComponents[ggg],Last])
Out[]=

Cycle Sizes for Simple vs. Random States [WRONG!]

NKS code

“Simple” initial condition is a single large blob [[[ we know there is a particular cycle that has the large blob case ; as well as its cyclic friends ]]]
Only one of these is left-right symmetric
Some of the longest lived are actually the most regular.....
Use weighted randomness, etc.

Size 30 random inits

Majority of randomly chosen states are on short cycles....
[[ Can we predict period from e.g. particle count? ]]

Size 80, 50, etc.