Major effects as edges are removed

The effective dimension can increase because removing a fixed fraction of edges has more effect on the interior of a geodesic than the exterior
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Range[10]
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{1,2,3,4,5,6,7,8,9,10}
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GridGraph[{10,10}]
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DiscretizeRegion[Rectangle[]]
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ResourceFunction["FlatManifoldToGraph"][2,.2,200]["SpatialGraph"]
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ResourceFunction["FlatManifoldToGraph"][2,.15,200]["SpatialGraph"]
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Graphics[Point[RandomPoint[Rectangle[],200]]]
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NearestNeighborGraph[RandomPoint[Rectangle[],200],{All,.15}]
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RandomGraph[{100,400}]
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Table[NearestNeighborGraph[RandomPoint[Rectangle[],200],{All,f}],{f,.05,.25,.05}]
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When we estimate dimension or curvature, we are computing V(r) .... we can do this with rmin < r < rmax
Analog of RipleyK for graph distance....
We need it to be above the percolation threshold where there is an infinite cluster with probability 1.

Percolation in graphs

The case of a torus graph

Effective mean field theory

Fraction f of edges have been removed....
Geodesic ball volume r^d
Assume it’s “manifold like” : i.e.
d[f]
d[t]

Dimension measurement