Meassure dimensionality of (hyper) graph

[freemin7]

The problem

In his presentation yesterday Steven Wolfram talked about the (fractional) dimensionality of hyper graphs. The documentation of this package does not show how calculate/plot them.

Possible solution

A example how to calculate those would be great. Note that these numbers depend a selected subgraph. A function that takes the new subgraph and repeatably does that calculation of dimensionality would be a quality of life feature

Alternative solutions

There is some extra package i don't know about.

Additional context

The end goal would generate plots like this: https://youtu.be/rbfFt2uNEyQ?t=1643

[GetJonWithIt]

https://resources.wolframcloud.com/FunctionRepository/resources/GraphNeighborhoodVolumes

Note, in general: https://www.wolframcloud.com/obj/wolframphysics/Tools/guide-page

-Jonathan

[freemin7]

Thank you for the pointer but https://resources.wolframcloud.com/FunctionRepository/resources/GraphNeighborhoodVolumes doesn't work for (Hyper) Graohs generated by SetReplace.

[maxitg]

There is also HypergraphNeighborhoodVolumes in Function Repository.

Also, if you go to the corresponding section in the Technical Introduction, you can click on the plot to copy the input (it takes a few minutes to compute in this case):

HypergraphDimensionEstimateList[hg_] := 
  ResourceFunction["LogDifferences"][
   MeanAround /@ 
    Transpose[
     Values[ResourceFunction["HypergraphNeighborhoodVolumes"][hg, All,
        Automatic]]]];
ListLinePlot[
 Select[Length[#] > 3 &][
  HypergraphDimensionEstimateList /@ 
   Drop[WolframModel[{{x, y}, {x, z}} -> {{x, z}, {x, w}, {y, w}, {z, 
        w}}, {{1, 2}, {1, 3}}, 16, "StatesList"], 4]], Frame -> True, 
 PlotStyle -> 
  WolframPhysicsProjectStyleData["GenericLinePlot", "PlotStyles"]]

(ResourceFunction["WolframModel"] and ResourceFunction["WolframPhysicsProjectStyleData"] call this package, but they introduce overhead, so I replaced them with the package functions in the above.)

[maxitg]

And also, yes, we do need a well-tested, easy-to-use function in SetReplace which would measure volumes and dimensionalities (including automatic averaging with error bars, sampling, etc.).

We also need a function that would color edges in the hypergraph itself based on the local dimensionality, so that we can see how it varies in space, probably as a special value for the PlotStyle option in WolframModelPlot: #307.

[freemin7]

What is the role of Drop and how to choose an appropriate value?

[maxitg]

Drop removes the first four states from the plot, because they are too small to measure dimension in a meaningful way, and just make the left-hand side of the plot too crowded. It actually does not change anything here because Select[Length[#] > 3 &] takes care of that anyway.

You can just use this code (3 here means you should have a hypergraph radius of at least 3, which is a reasonable minimum for any rule):

HypergraphDimensionEstimateList[hg_] := 
  ResourceFunction["LogDifferences"][
   MeanAround /@ 
    Transpose[
     Values[ResourceFunction["HypergraphNeighborhoodVolumes"][hg, All,
        Automatic]]]];
ListLinePlot[
 Select[Length[#] > 3 &][
  HypergraphDimensionEstimateList /@ 
   WolframModel[{{x, y}, {x, z}} -> {{x, z}, {x, w}, {y, w}, {z, 
       w}}, {{1, 2}, {1, 3}}, 16, "StatesList"]], Frame -> True, 
 PlotStyle -> 
  WolframPhysicsProjectStyleData["GenericLinePlot", "PlotStyles"]]