Dimensions fractales de réseaux vectoriels : méthodes d’estimation et robustesse des résultats

Abstract

Streams networks are part of transport networks and more generally of “spatial networks” that gave rise to fundamental researches (Barthelemy, 2011) and applied geography (Strano et al., 2012). Numerous studies have focused on fractal analysis of stream networks. However, only few papers compare or discuss the estimation methods and the uncertainty of the main fractal indicator, the fractal dimension. This work focuses on the fractal properties of vector networks both virtual and actual. We first mention the essential distinction between infinite mathematical fractal and nature fractal. Then, we present different theoretical and empirical dimensions that we use. In particular, we compare three fractal dimension estimators: the most classical estimator for stream networks, based on a topological approach with the Horton-Strahler ratios, and two other estimators based on a geometric approach, the box-counting dimension and the correlation dimension. Three main methodological results can be highlighted: 1- the study of virtual network contributes to the assessment of the various estimators’ relevance, according to the characteristics of networks; 2- an empirical fractal domain must be determined with an objective method to estimate fractal dimensions that can be compared; 3- the observation of uncertainty and stability of the fractal dimension is necessary for any valid comparison.

Keywords

• fractal dimension
• uncertainty
• stream networks
• virtual network