Abstract
Recently [L. Lacasa and J. Gómez-Gardeñes, Phys. Rev. Lett. 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 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 β of integer lattices ℤ(d) coincides with the Haussdorf dimension of their coarsely equivalent Euclidean spaces, β = d.
Highlights
During the last decade the science of networks has shed light on the importance that the real architecture of the interactions among the constituents of complex systems has on the onset of collective behavior [5,6,7]
We show that their correlation dimension coincides with the the Haussdorff dimension of the respective coarsely-equivalent Euclidean space
[1], we proposed an extension of the concept of correlation dimension [16] to estimate the dimensionality of complex networks by using random walkers to explore the network topology
Summary
During the last decade the science of networks has shed light on the importance that the real architecture of the interactions among the constituents of complex systems has on the onset of collective behavior [5,6,7]. [1], we proposed an extension of the concept of correlation dimension [16] to estimate the dimensionality of complex networks by using random walkers to explore the network topology. This extension builds up on the well-known Grassberger-Procaccia method [2,3,4], originally designed to quantify the fractal dimension of strange attractors in dissipative chaotic dynamical systems.
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