In this network every intersection represents a vertex. I know the degree of this network is 4 except for the corner vertices which have a much higher degree since a lot of curves start/end from those points. I'm trying to figure out the effective dimension of this network.
So apparently there's a simple test to determine the effective dimension of a network: Start from a node, then look at all nodes that are up to r connections away. If the network is behaving like it’s d-dimensional, then the number of nodes in that “ball” will be about r^d.
If the network behaves like flat d-dimensional space, then the number of nodes will always be close to r^d. But if it behaves like curved space, as in General Relativity, then there’s a correction term, that’s proportional to a mathematical object called the Ricci scalar.
How do you determine if the number of nodes is "about" r^d? That's not very exact wording. Intuitively this network looks like it might not behave like flat d-dimensional space but I'm not sure how to proceed to show this as fact. How does one determine the Ricci scalar if the network is indeed behaving like curved space? Lastly, what would be an interesting property to investigate with this network?
