cg[rule_,t_:20]:=ResourceFunction["WolframModel"][rule,Automatic,t,"LayeredCausalGraph"]
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spl[rule_,t_:20]:=ResourceFunction["WolframModel"][rule,Automatic,t,"StatesPlotsList"]
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mwcg[rule_,t_:5]:=ResourceFunction["MultiwaySystem"]["WolframModel"{rule},{First[rule]/._Integer0},t,"CausalGraphStructure"]
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mwbg[rule_,t_:5]:=ResourceFunction["MultiwaySystem"]["WolframModel"{rule},{First[rule]/._Integer0},t,"BranchialGraphStructure"]
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mw[rule_,t_:5]:=ResourceFunction["MultiwaySystem"]["WolframModel"{rule},{First[rule]/._Integer0},t,"StatesGraphStructure"]
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{{1,2},{1,3}}{{1,1},{1,4},{1,5}}

Single looped behavior

cg[{{1,1},{1,1}}{{1,1},{1,1},{1,2}}]
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mwcg[{{1,1},{1,1}}{{1,1},{1,1},{1,2}}]
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cg[{{1,1},{1,2}}{{1,1},{2,1},{2,3}}]
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mwcg[{{1,1},{1,2}}{{1,1},{2,1},{2,3}},6]
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SimpleGraph[%]
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Simple cosmological horizon

cg[{{1,1},{2,1}}{{1,1},{1,2},{3,1}}]
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mwcg[{{1,1},{2,1}}{{1,1},{1,2},{3,1}}]
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SimpleGraph[%]
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mwcg[{{1,1},{2,1}}{{1,1},{1,2},{3,1}},6]
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mwbg[{{1,1},{2,1}}{{1,1},{1,2},{3,1}},4]
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mwbg[{{1,1},{2,1}}{{1,1},{1,2},{3,1}},5]
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mw[{{1,1},{2,1}}{{1,1},{1,2},{3,1}},5]
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spl[{{1,1},{2,1}}{{1,1},{1,2},{3,1}},10]
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{{1,1},{1,2}}{{1,1},{1,2},{2,3}}
cg[{{1,1},{1,2}}{{2,2},{2,1},{1,3}},60]
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Partial horizon

cg[{{1,1},{1,2}}{{2,2},{2,1},{2,3}},60]
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cg[{{1,1},{1,2}}{{2,2},{2,1},{2,3}},80]
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Show[#,ImageSize{60,60}]&/@spl[{{1,1},{1,2}}{{2,2},{2,1},{2,3}},80]
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Turing machine like

cg[{{1,1},{1,2}}{{2,2},{2,3},{3,1}},40]
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spl[{{1,1},{1,2}}{{2,2},{2,3},{3,1}},5]
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mwcg[{{1,1},{1,2}}{{2,2},{2,3},{3,1}},10]
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mw[{{1,1},{1,2}}{{2,2},{2,3},{3,1}},10]
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Cosmological

cg[{{1,2},{1,3}}{{2,1},{1,3},{3,4}}]
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spl[{{1,2},{1,3}}{{2,1},{1,3},{3,4}}]
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Multitrack

cg[{{1,2},{1,3}}{{2,2},{3,2},{1,4}},30]
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cg[{{1,2},{1,3}}{{2,2},{1,3},{1,4}},40]
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cg[{{1,2},{1,3}}{{2,2},{2,4},{3,1}},30]
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mw[{{1,2},{1,3}}{{2,2},{2,4},{3,1}}]
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$Aborted
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cg[{{1,2},{2,3}}{{2,4},{2,3},{4,1}}]
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mwcg[{{1,2},{2,3}}{{2,4},{2,3},{4,1}},4]
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mw[{{1,2},{2,3}}{{2,4},{2,3},{4,1}},5]
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LayeredGraphPlot[%]
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cg[{{1,2},{3,2}}{{4,4},{2,4},{1,3}},30]
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mwcg[{{1,2},{3,2}}{{4,4},{2,4},{1,3}},5]
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mwcg[{{1,2},{3,2}}{{4,4},{2,4},{1,3}},6]
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$Aborted
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Terminating

cg[{{1,1},{1,2}}{{1,1},{2,3},{3,2}}]
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mwcg[{{1,1},{1,2}}{{1,1},{2,3},{3,2}}]
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cg[{{1,1},{1,2}}{{1,1},{2,3},{3,2}}]
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mwcg[{{1,1},{1,2}}{{1,1},{2,3},{3,2}}]
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cg[{{1,1},{1,2}}{{2,2},{3,2},{1,4}}]
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mwcg[{{1,1},{1,2}}{{2,2},{3,2},{1,4}}]
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cg[{{1,1},{1,2}}{{1,1},{2,3},{2,3}}]
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mwcg[{{1,1},{1,2}}{{1,1},{2,3},{2,3}}]
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cg[{{1,1},{1,2}}{{1,1},{2,3},{2,4}}]
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mwcg[{{1,1},{1,2}}{{1,1},{2,3},{2,4}}]
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cg[{{1,1},{1,2}}{{1,2},{1,2},{2,3}}]
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mwcg[{{1,1},{1,2}}{{1,2},{1,2},{2,3}}]
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cg[{{1,2},{2,3}}{{3,2},{3,2},{1,4}},20]
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mwcg[{{1,2},{2,3}}{{3,2},{3,2},{1,4}}]
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Other

cg[{{1,2},{2,3}}{{4,4},{3,4},{2,1}}]
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[ Slow to connect ]

cg[{{1,2},{2,3}}{{4,4},{4,3},{2,3}},30]
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spl[{{1,2},{2,3}}{{4,4},{4,3},{2,3}},5]
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cg[{{1,2},{2,3}}{{4,1},{1,5},{5,3}}]
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mwcg[{{1,2},{2,3}}{{4,1},{1,5},{5,3}},4]
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cg[{{1,2},{3,2}}{{1,3},{1,2},{4,3}},30]
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cg[{{1,2},{3,2}}{{2,2},{4,1},{1,3}},30]
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