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GridGraph[{30,30}]

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ResourceFunction["FlatManifoldToGraph"][10,1,100]["SpatialGraph"]

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In[]:=

ResourceFunction["FlatManifoldToGraph"][10,1,300]["SpatialGraph"]

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In[]:=

ResourceFunction["FlatManifoldToGraph"][10,.5,300]["SpatialGraph"]

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#### “Dimension diffusion”

“Dimension diffusion”

Fix the number of elements, and the number of connections....

In[]:=

ResourceFunction["WolframModel"][{{x,y},{y,z},{z,w},{w,v}}{{y,u},{u,v},{w,x},{x,u}},List@@@EdgeList@GridGraph[{10,10}],5,"StatesPlotsList"]

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ResourceFunction["WolframModel"][{{x,y},{y,z},{z,w},{w,v}}{{y,u},{u,v},{w,x},{x,u}},List@@@EdgeList@GridGraph[{5,5,5}],5,"StatesPlotsList"]

#### Given a graph, randomly delete edges; what does this do to the dimension?

Given a graph, randomly delete edges; what does this do to the dimension?

#### Is there a percolation-like phenomenon, e.g. a dimension phase transition?

Is there a percolation-like phenomenon, e.g. a dimension phase transition?

#### What do random changes to a graph do to dimension?

What do random changes to a graph do to dimension?

Might not go down, if you are effectively preferentially deleting from close to the point you’re starting from

Continuum theory of dimension change: what attributes of a graph does it depend on?

When you delete edges in a ball, the fraction of the periphery that is removed is much less than the fraction of the center that’s removed: i.e. there will be a higher dimension, not lower....

Probably the jumps are due to pieces of the universe breaking off.......

Looking out a small distance:

For the above radii, the “inside” is too small to get zapped ... so there is no increase in dimension....

#### Torus graph

Torus graph

There is no boundary .... but it never reaches its full “dimension potential” ;; there are lots of paths going around the universe