In section 5 we used the cone volume *C*_{t} to probe the large-scale structure of causal graphs generated by string substitution systems. Now we use *C*_{t} 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 *V*_{r} of balls in the hypergraphs generated by this rule grow roughly like *r*^{2.6}, 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 *V*_{r } and *C*_{t} 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}][[1]]}]

The linear increase in this plot implies exponential growth in *C*_{t} and indeed we find that for this rule:

This exponential growth—compared with the polynomial growth of *V*_{r}—implies 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 2_{2} 4_{2} globular-hypergraph-generating rules shown in the previous subsection show similar exponential growth in *C*_{t}, 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 *V*_{r} and *C*_{t} 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}][[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 *V*_{r} and *C*_{t} 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.