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Research done during the previous century established our Standard Cosmological Model. There are many details still to be filled in, but few would seriously doubt the basic premise. Past surveys have revealed that the large-scale…
Recently, a hybrid percolation transitions (HPT) that exhibits both a discontinuous transition and critical behavior at the same transition point has been observed in diverse complex systems. In spite of considerable effort to develop the…
Fractals are ubiquitous in nature, and since Mandelbrot's seminal insight into their structure, there has been growing interest in them. While the topological properties of the limit sets of IFSs have been studied -- notably in the…
The notion of the abundance of fractals is critically re-examined in light of surprising data regarding the scaling range in empirical reports on fractality.
In this article, we present a novel box-covering algorithm for analyzing the fractal properties of complex networks. Unlike traditional algorithms that impose a predetermined box size, our approach assigns nodes to boxes identified by their…
Compelling evidence exists to show that the structure of molecular clouds is fractal in nature. In this paper, the author reiterates this view and, in addition, asserts that not only is cloud geometry fractal, but that they also have a…
We review a range of stastistical methods for analyzing the structures of star clusters, and derive a new measure ${\cal Q}$ which both quantifies, and distinguishes between, a (relatively smooth) large-scale radial density gradient and…
There are many research papers dealing with fractal dimension of real-valued fractal functions in the recent literature. The main focus of the present paper is to study fractal dimension of complex-valued functions. This paper also…
We study long-range power-law correlated disorder on square and cubic lattices. In particular, we present high-precision results for the percolation thresholds and the fractal dimension of the largest clusters as function of the correlation…
We describe new families of random fractals, referred to as "V-variable", which are intermediate between the notions of deterministic and of standard random fractals. The parameter V describes the degree of "variability" : at each…
We develop an axiomatic framework for fractal analysis and fractal number theory grounded in hierarchies of definability. Central to this approach is a sequence of formal systems F_n, each corresponding to a definability level S_n contained…
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…
The study of human dynamics has attracted much interest from many fields recently. In this paper, the fractal characteristic of human behaviors is investigated from the perspective of time series constructed with the amount of library…
Analysis on fractals is a growing field, with hints of potential for widespread applicability across all of STEM. One of the most heavily researched type of fractals are the nested fractals, fractal shapes defined by virtue of being made of…
On the one hand, the large scale structure of matter is arguably scale invariant, and, on the other hand, halos and voids are recognized as prominent features of that structure. To unify both approaches, we propose to model the dark matter…
Fractional calculus is a generalization of classical theories of integration and differentiation to arbitrary order (i.e., real or complex numbers). In the last two decades, this new mathematical modeling approach has been widely used to…
Observations of galaxies over large distances reveal the possibility of a fractal distribution of their positions. The source of fractal behavior is the lack of a length scale in the two body gravitational interaction. However, even with…
The effect of geometry and morphology of superconducting structure on magnetic flux trapping is considered. It is found that the clusters of normal phase, which act as pinning centers, have significant fractal properties. The fractal…
We present a structural clustering algorithm for large-scale datasets of small labeled graphs, utilizing a frequent subgraph sampling strategy. A set of representatives provides an intuitive description of each cluster, supports the…
Identifying clusters of vertices in graphs continues to be an important problem, and modularity continues to be used as a tool for solving the problem. Modularity, which measures the quality of a division of the vertices into clusters,…