WOLFRAM NOTEBOOK

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ggg=Graph[Rule@@@WolframModel[{{0,1},{0,2},{0,3}}{{1,2},{2,3},{3,4},{4,1},{4,3}},Table[0,3,2],20,"FinalState"]]
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ResourceFunction["HypergraphToGraph"][WolframModel[{{0,1},{0,2},{0,3}}{{1,2},{2,3},{3,4},{4,1},{4,3}},Table[0,3,2],20,"FinalState"]]
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Histogram[LocalClusteringCoefficient[%60],"Log"]
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Histogram[ClosenessCentrality[%60]]
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HistogramFlattenGraphDistanceMatrix
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Undirected case:
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HistogramFlattenGraphDistanceMatrixUndirectedGraph@
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Histogram[Flatten[GraphDistanceMatrix[UndirectedGraph@ResourceFunction["HypergraphToGraph"][WolframModel[{{0,1},{0,2},{0,3}}{{1,2},{2,3},{3,4},{4,1},{4,3}},Table[0,3,2],25,"FinalState"]]]]]
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Functions on the graph

Start from a delta function at a given node; then relax for some number of steps
Then a way to compute distance between nodes is the extent of overlap of these distributions [ i.e. a softening of geodesic distance ]

Mean field theory

What is the probability that an event happens in a given part of the graph?
An event is analogous to e.g. collisions
If we had a bulk property of the graph that implies the probability for an event.......
In the strings case, consider a local entropy measure: or some other aggregate measure on n-tuples
We don’t know if a specific sequence will occur, but we’re making an aggregate statement about probabilities of n-tuples
Consider: ABB -> BA
First order mean field theory just has a probability of As and Bs pA pB^2
In a long string, pB could vary with position (but what is the position on a string??)
pB(x) : imagine string has unit length
E.g. Thue-Morse (“homogeneous”), or Cantor set (not homogeneous)
In a graph, it’s a probability for subgraphs
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