I am aware that curvature is a complicated thing and I am simplifying a lot. Also, we need to be very careful about what are we calling dimension. As we are considering a (hyper)graph model instead of a vector space we do not have a natural concept of dimension. But if this model is to represent the real world, then the actual dimensions we see everyday have to emerge from the model somehow. Studying the function of the growing of a ball is a cheap option for addressing it. Other option would be to study how many pairwise orthogonal geodesics can be incident on a single point, but that has the caveat of possible 'short' dimensions.
In those approaches above, and possibly any other, the metric used is fundamental. This is also hard is (hyper)graphs. The option of just counting edges in a path makes difficult to give place to the Euclidean-like metric of our everyday. Jonathan Gorard uses something with parallel transport that I do not quite understand yet (something about random walks pondered by a more basic distance I think).
Then, as you pointed to the Kaluza-Klein theory, I got thinking. We are supposed to add a fifth dimension, but dimensions are sorta meaningless in this context. We can force it by explicitly considering our graph to be some Cartesian product of other graphs. But the dimension also gives form to the growing function of the balls, so perhaps instead of trying to add an extra dimension it is better if we just try to relate its implications on the ball growing function with its implications on electromagnetism. And as Wolfram and Jonathan point, the r^n component is the 'principal' component of its size on a 'standard' n-dimensional space, and the r^(n+2) component is 'hinted' by the curvature. Of course that knowing this scalar component cannot give the full curvature tensor, and it is hard anyway to define such a tensor for a graph. But in a graph the balls do not need to grow as anr^n+a_(n+2)r^(n+2)+a(n+4)*r^(n+4)+O(r^(n+6)); we can also have a r^(n+1) term. As I have not read comment from Wolfram or Jonathan about this term I see possible for it to have some relevance.
When we have a time plus 3D space and add a short dimension for Kaluza's sake we get that balls grow as 4-dimensional at first, until the short dimension is completely 'absorbed' (not sure what word should I use for this); followed by the normal 3-dimensional growth. If we instead have a growing as r^4+r^5, it would be actually he other way around, since for large r, r^5 grows faster. But you also said that a small diameter is not necessary for Kaluza-Klein to apply, so I think this merits attention.