Major effects as edges are removed
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
In[]:=
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
Percolation in graphs
The case of a torus graph
The case of a torus graph
Effective mean field theory
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
Dimension measurement