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Color by Dimension...

In[]:=
gg14=Graph[Rule@@@WolframModel[{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}},{{1,2},{1,3}},14,"FinalState"]]
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gg12=Graph[Rule@@@WolframModel[{{x,y},{x,z}}{{x,z},{x,w},{y,w},{z,w}},{{1,2},{1,3}},12,"FinalState"]];
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ResourceFunction["GraphNeighborhoodVolumes"]
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ResourceFunction["GraphNeighborhoodVolumes"][UndirectedGraph[gg12]]
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Histogram[Values[Length/@ResourceFunction["GraphNeighborhoodVolumes"][UndirectedGraph[gg12]]]]
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ListLinePlot[Values[ResourceFunction["GraphNeighborhoodVolumes"][UndirectedGraph[gg12]]],Filling->Bottom,FillingStyleOpacity[.2]]
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$Aborted
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ListLinePlot[Take[Values[ResourceFunction["GraphNeighborhoodVolumes"][UndirectedGraph[gg12]]],20],Filling->Bottom,FillingStyleOpacity[.2]]
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ListLinePlot[RandomSample[Values[ResourceFunction["GraphNeighborhoodVolumes"][UndirectedGraph[gg12]]],40],Filling->Bottom,FillingStyleOpacity[.2]]
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Histogram[Values[ResourceFunction["GraphNeighborhoodVolumes"][UndirectedGraph[gg12]][[All,12]]]]
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gg12v12=Values[ResourceFunction["GraphNeighborhoodVolumes"][UndirectedGraph[gg12]][[All,12]]];
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Graph[gg12,VertexSizeThread[VertexList[gg12]gg12v12/50],PlotTheme"Default"]
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Graph[gg12,VertexSizeThread[VertexList[gg12]gg12v12/80],PlotTheme"Default"]
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