#### 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]

Out[]=

{1,2,3,4,5,6,7,8,9,10}

In[]:=

GridGraph[{10,10}]

Out[]=

In[]:=

DiscretizeRegion[Rectangle[]]

Out[]=

In[]:=

ResourceFunction["FlatManifoldToGraph"][2,.2,200]["SpatialGraph"]

Out[]=

In[]:=

ResourceFunction["FlatManifoldToGraph"][2,.15,200]["SpatialGraph"]

Out[]=

In[]:=

Graphics[Point[RandomPoint[Rectangle[],200]]]

Out[]=

In[]:=

NearestNeighborGraph[RandomPoint[Rectangle[],200],{All,.15}]

Out[]=

In[]:=

RandomGraph[{100,400}]

Out[]=

In[]:=

Table[NearestNeighborGraph[RandomPoint[Rectangle[],200],{All,f}],{f,.05,.25,.05}]

Out[]=

,

,

,

,

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....

In[]:=

ValuesResourceFunction["WolframHausdorffDimension"]

,All,"AllVolumes","VertexMethod"Identity,"VolumeMethod"Identity

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