# 6.8 Large-Scale Structure of Causal Graphs

In section 5 we used the cone volume Ct to probe the large-scale structure of causal graphs generated by string substitution systems. Now we use Ct to probe the large-scale structure of causal graphs generated by our models.

Consider for example the rule

{{x, y}, {x, z}} -> {{x, y}, {x, w}, {y, w}, {z, w}}

We found in section 4 that after a few steps, the volumes Vr of balls in the hypergraphs generated by this rule grow roughly like r2.6, suggesting that in the limit the hypergraphs behave like a finite-dimensional space, with dimension 2.6.

The pictures below show the log differences in Vr and Ct for this rule after 15 steps of evolution:

With[{w = ResourceFunction[ "WolframModel"][{{x, y}, {x, z}} -> {{x, z}, {x, w}, {y, w}, {z, w}}, {{0, 0}, {0, 0}}, 15]}, ListLinePlot[ResourceFunction["LogDifferences"][#], Frame -> True, PlotStyle -> {Hue[0.985, 0.845, 0.638], Hue[0.05, 0.949, 0.955], Hue[0.089, 0.75, 0.873], Hue[0.06, 1., 0.8], Hue[0.12, 1., 0.9], Hue[0.08, 1., 1.], Hue[0.987, 0.673, 0.733], Hue[ 0.04, 0.68, 0.94], Hue[0.995, 0.989, 0.824], Hue[ 0.991, 0.4, 0.9]}] & /@ {ResourceFunction["RaggedMeanAround"][ Values@ResourceFunction["HypergraphNeighborhoodVolumes"][ w["FinalState"]]], ResourceFunction["GraphNeighborhoodVolumes"][ w["CausalGraph"], {1}][]}]

The linear increase in this plot implies exponential growth in Ct and indeed we find that for this rule: This exponential growthcompared with the polynomial growth of Vrimplies that expansion according to this rule is in a sense sufficiently rapid that there is increasing causal disconnection between different parts of the system.

The other three 22 42 globular-hypergraph-generating rules shown in the previous subsection show similar exponential growth in Ct, at least over the number of steps of evolution tested.

A rule such as

{{x, y, y}, {x, z, u}} -> {{u, v, v}, {v, z, y}, {x, y, v}}

whose hypergraph and causal graph (after 500 steps) are respectively

With[{w = ResourceFunction[ "WolframModel"][{{x, y, y}, {x, z, u}} -> {{u, v, v}, {v, z, y}, {x, y, v}}, {{0, 0, 0}, {0, 0, 0}}, 500]}, {ResourceFunction["WolframModelPlot"][w["FinalState"]], w["CausalGraph"]}]

gives the following for the log differences of Vr and Ct after 10,000 steps:

With[{w = ResourceFunction[ "WolframModel"][{{x, y, y}, {x, z, u}} -> {{u, v, v}, {v, z, y}, {x, y, v}}, {{0, 0, 0}, {0, 0, 0}}, 10000]}, ListLinePlot[ResourceFunction["LogDifferences"][#], Frame -> True, PlotRange -> {0, 2.5}, PlotStyle -> ResourceFunction["WolframPhysicsProjectStyleData"][ "GenericLinePlot", "PlotStyles"]] & /@ {ResourceFunction[ "RaggedMeanAround"][ Values@ResourceFunction["HypergraphNeighborhoodVolumes"][ w["FinalState"]]], ResourceFunction["GraphNeighborhoodVolumes"][ w["CausalGraph"], {1}][]}]

This implies that for this rule the hypergraphs it generates and its causal graph both effectively limit to finite-dimensional spaces, with the hypergraphs having dimension perhaps slightly over 2, and the causal graph having dimension 2.

Consider now the rule:

{{x, y, x}, {x, z, u}} -> {{u, v, u}, {v, u, z}, {x, y, v}}

The hypergraph and causal graph (after 1500 steps) for this rule are respectively:

With[{w = ResourceFunction[ "WolframModel"][{{x, y, x}, {x, z, u}} -> {{u, v, u}, {v, u, z}, {x, y, v}}, {{0, 0, 0}, {0, 0, 0}}, 1500]}, {ResourceFunction["WolframModelPlot"][w["FinalState"]], w["CausalGraph"]}]

The log differences of Vr and Ct after 10,000 steps are then:

With[{w = ResourceFunction[ "WolframModel"][{{x, y, x}, {x, z, u}} -> {{u, v, u}, {v, u, z}, {x, y, v}}, {{0, 0, 0}, {0, 0, 0}}, 10000]}, ListLinePlot[ResourceFunction["LogDifferences"][#], Frame -> True, PlotRange -> {0, 3.5}, PlotStyle -> ResourceFunction["WolframPhysicsProjectStyleData"][ "GenericLinePlot", "PlotStyles"]] & /@ {ResourceFunction[ "RaggedMeanAround"][ Values@ResourceFunction["HypergraphNeighborhoodVolumes"][ w["FinalState"]]], ResourceFunction["RaggedMeanAround"][ Values@ResourceFunction["GraphNeighborhoodVolumes"][ w["CausalGraph"]]]}]

Both suggest limiting spaces with dimension 2, but with a certain amount of (negative) curvature.