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A novel fractal analysis of the cosmic web structure is carried out, employing the Sloan Digital Sky Survey, data release 7. We consider the large-scale stellar mass distribution, unlike other analyses, and determine its multifractal…
Human settlements on Earth are scattered in a multitude of shapes, sizes and spatial arrangements. These patterns are often not random but a result of complex geographical, cultural, economic and historical processes that have profound…
Fractal geometry, defined by self-similar patterns across scales, is crucial for understanding natural structures. This work addresses the fractal inverse problem, which involves extracting fractal codes from images to explain these…
Passenger flows in a traffic network reflect spatial interaction patterns in an urban systems. Gravity models can be employed to quantitatively describe and predict spatial flows. However, how to model passenger flows and reveal the deep…
Many aggregate distributions of urban activities such as city sizes reveal scaling but hardly any work exists on the properties of spatial distributions within individual cities, notwithstanding considerable knowledge about their fractal…
This thesis focuses on characterizing the distribution of points and galaxies using multifractal analysis. In this attempt the main emphasis is on calculating the Minkowski-Bouligand fractal dimension (Dq) of the distribution of points over…
A time series of fractal dimension values of urban form can form a fractal dimension curve and reflects urban growth. In many cases, the fractal dimension curves of cities can be modeled with logistic function, which in turn can be used to…
Simple time-reversible systems can generate {\it irreversible} flows satisfying the Second Law of Thermodynamics. Maps, and equivalent random walks, can also do this. We study a pair of time-reversible Baker Maps, $N2$ and $N3$, which…
Urban populations exhibit fractal organization and systematic scaling regularities, yet the scaling exponents reported across cities vary substantially, challenging existing theory. Using 100~m gridded population maps for 477 urban areas…
Numerous entropy-type characteristics (functionals) generalizing R\'enyi entropy are widely used in mathematical statistics, physics, information theory, and signal processing for characterizing uncertainty in probability distributions and…
We develop information-theoretic measures of spatial structure and pattern in more than one dimension. As is well known, the entropy density of a two-dimensional configuration can be efficiently and accurately estimated via a converging…
Fractal-like structures of varying complexity are common in nature, and measure-based dimensions (Minkowski, Hausdorff) supply their basic geometric characterization. However, at the level of fundamental dynamics, which is quantum,…
Dimension reduction, widely used in science, maps high-dimensional data into low-dimensional space. We investigate a basic mathematical model underlying the techniques of stochastic neighborhood embedding (SNE) and its popular variant…
The structure of the large scale distribution of the galaxies have been widely studied since the publication of the first catalogs. Since large redshift samples are available, their analyses seem to show fractal correlations up to the…
While the universe becomes more and more homogeneous at large scales, statistical analysis of galaxy catalogs have revealed a fractal structure at small-scales (\lambda < 100 h^{-1} Mpc), with a fractal dimension D=1.5-2 (Sylos Labini et al…
The family of visibility algorithms were recently introduced as mappings between time series and graphs. Here we extend this method to characterize spatially extended data structures by mapping scalar fields of arbitrary dimension into…
Cities are different around the world, but does this fact have any relation to culture? The idea that urban form embodies idiosyncrasies related to cultural identities captures the imagination of many in urban studies, but it is an…
Many models of fractal growth patterns (like Diffusion Limited Aggregation and Dielectric Breakdown Models) combine complex geometry with randomness; this double difficulty is a stumbling block to their elucidation. In this paper we…
We present a review of the history and the present state of the fractal approach to the large-scale distribution of galaxies. Angular correlation function was used as a general instrument for the structure analysis. It was realized later…
Precise analyses of the statistical and scaling properties of galaxy distribution are essential to elucidate the large-scale structure of the universe. Given the ongoing debate on its statistical features, the development of statistical…