Tuples[{1,-1},5]
In[]:=
Out[]=
odm=DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot]
In[]:=
Out[]=
MatrixPlot[%]
In[]:=
Out[]=
hg=Graph[Position[DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot],1]]
In[]:=
Out[]=
hdm=GraphDistanceMatrix[hg]
In[]:=
Out[]=
MatrixPlot[%]
In[]:=
Out[]=
odm-hdm
In[]:=
Out[]=
EdgeCount[%127]
In[]:=
640
Out[]=

Case of size 3

Tuples[{1,-1},3]
In[]:=
{{1,1,1},{1,1,-1},{1,-1,1},{1,-1,-1},{-1,1,1},{-1,1,-1},{-1,-1,1},{-1,-1,-1}}
Out[]=
DistanceMatrix[Tuples[{1,-1},3],DistanceFunctionDot]
In[]:=
{{3,1,1,1,1,1,1,3},{1,3,1,1,1,1,3,1},{1,1,3,1,1,3,1,1},{1,1,1,3,3,1,1,1},{1,1,1,3,3,1,1,1},{1,1,3,1,1,3,1,1},{1,3,1,1,1,1,3,1},{3,1,1,1,1,1,1,3}}
Out[]=
Position[%,1]
In[]:=
Out[]=
Graph[%]
In[]:=
Out[]=
SimpleGraph[%]
In[]:=
Out[]=
GraphDistanceMatrix

In[]:=
{{0,1,1,1,1,1,1,2},{1,0,1,1,1,1,2,1},{1,1,0,1,1,2,1,1},{1,1,1,0,2,1,1,1},{1,1,1,2,0,1,1,1},{1,1,2,1,1,0,1,1},{1,2,1,1,1,1,0,1},{2,1,1,1,1,1,1,0}}
Out[]=
%135
In[]:=
{{3,1,1,1,1,1,1,3},{1,3,1,1,1,1,3,1},{1,1,3,1,1,3,1,1},{1,1,1,3,3,1,1,1},{1,1,1,3,3,1,1,1},{1,1,3,1,1,3,1,1},{1,3,1,1,1,1,3,1},{3,1,1,1,1,1,1,3}}
Out[]=

Transitivity

(First/@Gather[ParallelMapMonitored[Catch[Module[{u},Do[u=MultiwaySystem[#,"A",t,"BranchialGraphStructure"];If[VertexCount[u]>150||EdgeCount[u]>150,Throw[u]],{t,9}];u]]#&,Catenate[allrules]],IsomorphicGraphQ[First[#],First[#2]]&])
In[]:=
Out[]=
TransitiveClosureGraph

In[]:=
FindClique

In[]:=
{{AAAAAAAAAAB,AAAAAAABB,AAAAAABAB,AAAAABAAB,AAAABAAAB,AAABAAAAB,AABAAAAAB,ABAAAAAAB,BAAAAAAAB}}
Out[]=
FindClique
,8,All
In[]:=
Out[]=
FindClique
,{7},All
In[]:=
{{AABAAAAAB,BAAAAAAAB,BAAAABB,BAAABAB,BAABAAB,BABAAAB,BBAAAAB},{AAABAAAAB,ABAAAAAAB,ABAAABB,ABAABAB,ABABAAB,ABBAAAB,BBAAAAB},{AAAAAABAB,AAAABAAAB,AAAABBB,AAABBAB,AABABAB,ABAABAB,BAAABAB},{AAAAAAABB,AAAAABAAB,AAAABBB,AAABABB,AABAABB,ABAAABB,BAAAABB},{AAAAABAAB,AAABAAAAB,AAABABB,AAABBAB,AABBAAB,ABABAAB,BAABAAB},{AAAABAAAB,AABAAAAAB,AABAABB,AABABAB,AABBAAB,ABBAAAB,BABAAAB}}
Out[]=
HighlightGraph
,%
In[]:=
Out[]=

Models of Hilbert Space

Graphics3D[Point[Tuples[{-1,1},3]]]
In[]:=
Out[]=
Tuples[{-1,1},5]
In[]:=
Out[]=
GridGraph[Table[2,5]]
In[]:=
Out[]=
Position[DistanceMatrix[Tuples[{-1,1},5],DistanceFunction(Norm[#1-#2]^2&)],4]
In[]:=
Out[]=
Graph[%]
In[]:=
Out[]=
SimpleGraph[%]
In[]:=
Out[]=
GlobalClusteringCoefficient

In[]:=
0
Out[]=
RaggedMeanAroundValuesGraphNeighborhoodVolumes

In[]:=
{1,6,16,26,31,32}
Out[]=
Differences[%]
In[]:=
{5,10,10,5,1}
Out[]=
Table[First@Values[GraphNeighborhoodVolumes[GridGraph[Table[2,n]],{1},Automatic]],{n,10}]
In[]:=
{{1,2},{1,3,4},{1,4,7,8},{1,5,11,15,16},{1,6,16,26,31,32},{1,7,22,42,57,63,64},{1,8,29,64,99,120,127,128},{1,9,37,93,163,219,247,255,256},{1,10,46,130,256,382,466,502,511,512},{1,11,56,176,386,638,848,968,1013,1023,1024}}
Out[]=
https://oeis.org/A008949
ListLinePlot[%151]
In[]:=
Out[]=
ListLinePlot[Table[First@Values[GraphNeighborhoodVolumes[GridGraph[Table[2,n]],{1},Automatic]]/2^n,{n,10}],PlotRangeAll]
In[]:=
Out[]=
ListLinePlot[Log[10,%151]]
In[]:=
Out[]=
Ternary n-cube:
Table[First@Values[GraphNeighborhoodVolumes[GridGraph[Table[3,n]],{1},Automatic]],{n,10}]
In[]:=
Out[]=
ListLinePlot[%]
In[]:=
Out[]=
ListLinePlot[Log[10,%158]]
In[]:=
Out[]=
T(n, 0) = 1, T(n, n) = 2^n, T(n, k) = T(n-1, k-1) + T(n-1, k)
Accumulate/@Table[Binomial[n,i],{n,0,20},{i,0,n}]
In[]:=
Out[]=
Table[First@Values[GraphNeighborhoodVolumes[GridGraph[Table[2,n]],{1},Automatic]],{n,10}]
In[]:=
{{1,2},{1,3,4},{1,4,7,8},{1,5,11,15,16},{1,6,16,26,31,32},{1,7,22,42,57,63,64},{1,8,29,64,99,120,127,128},{1,9,37,93,163,219,247,255,256},{1,10,46,130,256,382,466,502,511,512},{1,11,56,176,386,638,848,968,1013,1023,1024}}
Out[]=

This distance was Euclidean distance....

But in Hilbert space, distance is dot product....

DistanceMatrix[Tuples[{-1,1},5],DistanceFunction(Norm[#1-#2]^2&)]//ArrayPlot
In[]:=
Out[]=
Table[ArrayPlot[DistanceMatrix[Tuples[{0,1},n],DistanceFunction(Norm[#1-#2]^2&)],ImageSizeTiny],{n,1,6}]
In[]:=
Out[]=
GraphDistanceMatrix[GridGraph[Table[2,4]]]//MatrixPlot
In[]:=
Out[]=
DistanceMatrix[Tuples[{0,1},3],DistanceFunction(Norm[#1-#2]^2&)]
In[]:=
{{0,1,1,2,1,2,2,3},{1,0,2,1,2,1,3,2},{1,2,0,1,2,3,1,2},{2,1,1,0,3,2,2,1},{1,2,2,3,0,1,1,2},{2,1,3,2,1,0,2,1},{2,3,1,2,1,2,0,1},{3,2,2,1,2,1,1,0}}
Out[]=
GraphDistanceMatrix[GridGraph[Table[2,3]]]
In[]:=
{{0,1,1,2,1,2,2,3},{1,0,2,1,2,1,3,2},{1,2,0,1,2,3,1,2},{2,1,1,0,3,2,2,1},{1,2,2,3,0,1,1,2},{2,1,3,2,1,0,2,1},{2,3,1,2,1,2,0,1},{3,2,2,1,2,1,1,0}}
Out[]=

Comparisons

ParallelTable[RaggedMeanAround[Values[GraphNeighborhoodVolumes[MultiwaySystem[{"""A","""B"},"A",t,"BranchialGraphStructure"]]]],{t,2,10}]
In[]:=
Out[]=
ListLinePlot[Log[10,%]]
In[]:=
Out[]=

Finite Hilbert space


Approximating Hilbert Space [WRONG]

odm=(5-DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot])/2
In[]:=
Out[]=
MatrixPlot[%]
In[]:=
Out[]=
SimpleGraph[Position[(5-DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot])/2,1]]
In[]:=
Out[]=
GraphDistanceMatrix[%]
In[]:=
Out[]=
MatrixPlot[%]
In[]:=
Out[]=
%168-%172
In[]:=
Out[]=
RaggedMeanAround[Values[GraphNeighborhoodVolumes[SimpleGraph[Position[(5-DistanceMatrix[Tuples[{1,-1},5],DistanceFunctionDot])/2,1]]]]]
In[]:=
{1,11,32}
Out[]=
Table[RaggedMeanAround[Values[GraphNeighborhoodVolumes[SimpleGraph[Position[(n-DistanceMatrix[Tuples[{1,-1},n],DistanceFunctionDot])/2,1]]]]],{n,2,10}]
In[]:=
{{1,3,4},{1,7,8},{1,9,16},{1,11,32},{1,13,44,64},{1,15,58,128},{1,17,74,186,256},{1,19,92,260,512},{1,21,112,352,772,1024}}
Out[]=
N[Ratios[#]]&/@%
In[]:=
{{3.,1.33333},{7.,1.14286},{9.,1.77778},{11.,2.90909},{13.,3.38462,1.45455},{15.,3.86667,2.2069},{17.,4.35294,2.51351,1.37634},{19.,4.84211,2.82609,1.96923},{21.,5.33333,3.14286,2.19318,1.32642}}
Out[]=
ListLinePlot[%178]
In[]:=
Out[]=

Distance 1 apart is like sphere packing....

E8 lattice ?

Next try

Tuples[{1,-1},4]
In[]:=
Out[]=
DistanceMatrix[Tuples[{1,-1},4],DistanceFunction

Non-Flat Hilbert Space

Projective Hilbert space

Mod out by projective transformation
GridGraph[Table[3,5]]
In[]:=
Out[]=
GridGraph[Table[2,5]]
In[]:=
Out[]=
GridGraph[Table[2,2]]
In[]:=
Out[]=
GridGraph[Table[5,2]]
In[]:=
Out[]=

There is a finite-dimensional projective Hilbert space, that has a Kahler metric (CP^n)

SimpleGraph[TorusGraph[Table[3,5]]]
In[]:=
Out[]=
RaggedMeanAround[Values[GraphNeighborhoodVolumes[%]]]
In[]:=
{1,11,51,131,211,243}
Out[]=
Accumulate/@Table[Binomial[n,i],{n,0,10},{i,0,n}]
In[]:=
{{1},{1,2},{1,3,4},{1,4,7,8},{1,5,11,15,16},{1,6,16,26,31,32},{1,7,22,42,57,63,64},{1,8,29,64,99,120,127,128},{1,9,37,93,163,219,247,255,256},{1,10,46,130,256,382,466,502,511,512},{1,11,56,176,386,638,848,968,1013,1023,1024}}
Out[]=
Table[Graph[TorusGraph[Table[3,n]],ImageSizeTiny],{n,4}]
In[]:=
Out[]=
GraphDiameter

In[]:=
4
Out[]=
RaggedMeanAround[Values[GraphNeighborhoodVolumes[#]]]&/@%
In[]:=
{{1,3},{1,5,9},{1,7,19,27},{1,9,33,65,81}}
Out[]=
Ratios/@%//N
In[]:=
{{3.},{5.,1.8},{7.,2.71429,1.42105},{9.,3.66667,1.9697,1.24615}}
Out[]=

​