Analytical estimation of the correlation dimension of integer lattices

In this article we address the concept of correlation dimension which has been recently extended to network theory in order to eciently characterize and estimate the dimensionality and geometry of complex networks [1]. This extension is inspired in the Grassberger-Procaccia method [2{4], originally designed to quantify the fractal dimension of strange attractors in dissipative chaotic dynamical systems. When applied to networks, it proceeds by capturing the trajectory of a random walker di using over a network with well de ned dimensionality. From this trajectory, an estimation of the network correlation dimension is retrieved by looking at the scaling of the walker's correlation integral. Here we give analytical support to this methodology by obtaining the correlation dimension of synthetic networks representing well-de ned limits of real networks. In particular, we explore fully connected networks and integer lattices, these latter being coarsely-equivalent [20] to Euclidean spaces. We show that their correlation dimension coincides with the the Haussdor dimension of the respective coarsely-equivalent Euclidean space