A Class of Models with the Potential to Represent Fundamental Physics
  1. Introduction
  2. Basic Form of Models
  3. Typical Behaviors
  4. Limiting Behavior and Emergent Geometry
  5. The Updating Process for String Substitution Systems
  6. The Updating Process in Our Models
  7. Equivalence and Computation in Our Models
  8. Potential Relation to Physics
  9. Additional Material
  10. References
  11. Index

2.5 Multiedges

Rules can involve several copies of the same relation, corresponding to multiedges in a graph. A simple example is the rule:

{{x, y}} -> {{x, z}, {x, z}, {y, z}}
RulePlot[ResourceFunction[ "WolframModel"][{{x, y}} -> {{x, z}, {x, z}, {y, z}}], VertexLabels -> Automatic, "RulePartsAspectRatio" -> 0.25]

Running this rule produces a structure with 3n edges and  nodes at step n:

ResourceFunction[ "WolframModel"][{{{x, y}} -> {{x, z}, {x, z}, {y, z}}}, {{1, 1}}, 5]["StatesPlotsList", "MaxImageSize" -> 180]

Rules can both create and destroy multiedges. The rule

{{x, y}} -> {{x, z}, {z, w}, {y, z}}
RulePlot[ResourceFunction[ "WolframModel"][{{x, y}} -> {{x, z}, {y, z}, {z, w}}], VertexLabels -> Automatic, "RulePartsAspectRatio" -> 1]

generates a multiedge after one step, but then destroys it:

ResourceFunction[ "WolframModel"][{{{x, y}} -> {{x, z}, {z, w}, {y, z}}}, {{1, 1}}, 5]["StatesPlotsList", "MaxImageSize" -> 180]