The Correlation Dimension of a Rectilinear Grid
The correlation dimension dC of a finite network measures how the fraction of nodes at a given distance from a random node scales with the distance. However, there is no standardized formal definition of dC for a network. We consider various possible definitions of dC for a finite unweighted and undirected rectilinear grid in one, two, and three dimensions. We propose a simple “overall slope” definition for dC which yields an exact closed form expression for such grids. We prove that the overall slope definition satisfies two properties that should be satisfied by any definition of the correlation dimension of a network. Lastly, we present a conjecture giving a closed form expression for the overall slope correlation dimension of a finite rectilinear grid in E dimensions, for any positive integer E.