Analytical estimation of the correlation dimension of integer lattices

TL;DR

This paper analytically estimates the correlation dimension of integer lattices, confirming that it matches the Euclidean Hausdorff dimension, thus providing theoretical support for a recently proposed network fractal measure.

Contribution

It provides the first analytical derivation of the correlation dimension for integer lattices, validating previous numerical findings and linking it to classical geometric dimensions.

Findings

Correlation dimension of integer lattices equals their Euclidean dimension.

Analytical results agree with previous numerical estimates.

Supports the use of correlation dimension as a geometric measure for networks.

Abstract

Recently [L. Lacasa and J. G\'omez-Garde\~nes, Phys. Rev. Lett. {\bf 110}, 168703 (2013)], a fractal dimension has been proposed to characterize the geometric structure of networks. This measure is an extension to graphs of the so called {\em correlation dimension}, originally proposed by Grassberger and Procaccia to describe the geometry of strange attractors in dissipative chaotic systems. The calculation of the correlation dimension of a graph is based on the local information retrieved from a random walker navigating the network. In this contribution we study such quantity for some limiting synthetic spatial networks and obtain analytical results on agreement with the previously reported numerics. In particular, we show that up to first order the correlation dimension $\beta$ of integer lattices $\mathbb{Z}^d$ coincides with the Haussdorf dimension of their coarsely-equivalent Euclidean spaces, $\beta=d$.