Universe 8141 from The Wolfram Physics Project

5 steps

wm8141`wm8141`

signature

1₄→3₄

rule

{{{1, 1, 2, 3}} -> {{2, 2, 4, 4}, {4, 4, 1, 3}, {3, 2, 5, 1}}}

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Basic Evolution

Basic evolution:

[◼]	WolframModel

[{{{1,1,2,3}}{{2,2,4,4},{4,4,1,3},{3,2,5,1}}},{{1,1,1,1}},5,"StatesPlotsList"]

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

Event-by-event evolution:

[◼]	WolframModel

[{{{1,1,2,3}}{{2,2,4,4},{4,4,1,3},{3,2,5,1}}},{{1,1,1,1}},<|"MaxEvents"6|>,"EventsStatesPlotsList"]

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

Vertex and edge counts:

{vertexCountList,edgeCountList}=

[◼]	WolframModel

[{{{1,1,2,3}}{{2,2,4,4},{4,4,1,3},{3,2,5,1}}},{{1,1,1,1}},12,{"VertexCountList","EdgeCountList"}];

ListLogPlot{vertexCountList,edgeCountList},



	vertices
	edges

Symbolic expression for vertex count:

FindSequenceFunction[vertexCountList,t]

DifferenceRootFunction{y.,n.},-6n.+(2-n.)y.[n.]+(2-2n.)y.[1+n.]+(-6+n.)y.[2+n.]+2y.[3+n.]0,y.[1]1,y.[2]3,y.[3]9[t]

Symbolic expression for edge count:

FindSequenceFunction[edgeCountList,t]

DifferenceRootFunction{y.,n.},-6n.+(2-n.)y.[n.]+(2-2n.)y.[1+n.]+(-6+n.)y.[2+n.]+2y.[3+n.]0,y.[1]1,y.[2]3,y.[3]9[t]

Result after

generations:

WolframModel[

]

["FinalStatePlot"]

Causal Graph

Causal graph:

WolframModel[

]

"CausalGraph",

Rule[

]



Layered rendering:

WolframModel[

]

["LayeredCausalGraph"]

Causal graph distance matrix:

MatrixPlotTransposeGraphDistanceMatrix

WolframModel[

]

["CausalGraph"],



Final State Properties

Hypergraph adjacency matrix:

MatrixPlotAdjacencyMatrix@CatenateMapUndirectedEdge@@@Subsets[#,{2}]&,

WolframModel[

]

["FinalState"],



Vertex degree distribution:

HistogramValuesCountsCatenateUnion/@

WolframModel[

]

["FinalState"],



Neighborhood volumes (ignoring directedness of connections):

volumes=

[◼]	RaggedMeanAround

Values

[◼]	HypergraphNeighborhoodVolumes



WolframModel[

]

["FinalState"],All,Automatic;

ListLogLogPlotvolumes,



Effective dimension versus radius:

ListLinePlot

[◼]	LogDifferences

[volumes],



Successive neighborhood balls around a random vertex:

[◼]	HypergraphNeighborhoods



WolframModel[

]

["FinalState"],4

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

Distance matrix:

distanceMatrix=GraphDistanceMatrixUndirectedGraph

[◼]	HypergraphToGraph



WolframModel[

]

["FinalState"];

MatrixPlotExp[-(distanceMatrix/.0None)],



Distribution of distances in the graph:

HistogramFlatten[distanceMatrix],



Spreading of Effects

Causal graph adjacency matrix:

MatrixPlotAdjacencyMatrix

WolframModel[

]

["CausalGraph"],



Neighborhood volumes in causal graph:

ListLogLogPlotValues

[◼]	GraphNeighborhoodVolumes



WolframModel[

]

["CausalGraph"],{1},



Other Evolution Orders

Random evolutions:

[◼]	WolframModel

[{{{1,1,2,3}}{{2,2,4,4},{4,4,1,3},{3,2,5,1}}},{{1,1,1,1}},<|"MaxEvents"80|>,"FinalStatePlot","EventOrderingFunction""Random"]

Different deterministic evolution orders:

[◼]	WolframModel

[{{{1,1,2,3}}{{2,2,4,4},{4,4,1,3},{3,2,5,1}}},{{1,1,1,1}},<|"MaxEvents"80|>,"EventOrderingFunction"{#,"LeastRecentEdge","RuleOrdering","RuleIndex"}]["FinalStatePlot",PlotLabel#]&/@{"OldestEdge","LeastOldEdge","LeastRecentEdge","NewestEdge","RuleOrdering","ReverseRuleOrdering"}

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