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PaddedGraphNeighborhoodVolumes[graph_?GraphQ,verts_:All]:=Module{vertices=verts,rmax=Infinity},Switch[vertices,All,vertices=VertexList[graph],"Random"n_Integer,vertices=RandomSample[VertexList[graph],UpTo[Last[vertices]]]];Withres=AssociationParallelMapModule vertexCounter=0, lastDistance=-1, #AppendIf[#==={},#,Rest[First[#]]]&@Reap[Catch[BreadthFirstScan[ graph, #, "DiscoverVertex"If[#3>rmax,Throw[0,"rmax"]];If[#3>lastDistance,lastDistance=#3;Sow[vertexCounter]]; vertexCounter++&],"rmax"]]2,vertexCounter&,vertices,With[{max=Max[Length/@res]},PadRight[#,max,Last[#]]&/@res]/;ListQ[vertices]

(*ForStephentorun-~10%samplesize,5samples*)

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leanresults=Values[PaddedGraphNeighborhoodVolumes[theoremgraph,"Random"100]]

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ListLinePlot[MeanAround/@Transpose[%380]]

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uleanresults=Values[PaddedGraphNeighborhoodVolumes[UndirectedGraph[theoremgraph],"Random"100]];

ListLinePlot[MeanAround/@Transpose[%]]

ParallelTable[ListLinePlot[ResourceFunction["LogDifferences"][MeanAround/@Transpose[Values[PaddedGraphNeighborhoodVolumes[theoremgraph,"Random"3500]]]]],{5}]

ParallelTable[ListLinePlot[ResourceFunction["LogDifferences"][MeanAround/@Transpose[Values[PaddedGraphNeighborhoodVolumes[metamathGraph,"Random"3500]]]]],{5}]