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Title: Analytical estimation of the correlation dimension of integer lattices

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{sup d} coincides with the Haussdorf dimension of their coarsely equivalent Euclidean spaces, β = d.
Authors:
 [1] ;  [2] ;  [3]
  1. School of Mathematical Sciences, Queen Mary University of London, Mile End Road, E14NS London (United Kingdom)
  2. Institute for Biocomputation and Physics of Complex System (BIFI), Universidad de Zaragoza, Zaragoza (Spain)
  3. (Spain)
Publication Date:
OSTI Identifier:
22350999
Resource Type:
Journal Article
Resource Relation:
Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 24; Journal Issue: 4; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ATTRACTORS; CHAOS THEORY; CORRELATIONS; DIAGRAMS; EUCLIDEAN SPACE; FRACTALS; GEOMETRY; GRAPH THEORY