A Universal Rank-Size Law

A mere hyperbolic law, like the Zipf’s law power function, is often inadequate to describe rank-size relationships. An alternative theoretical distribution is proposed based on theoretical physics arguments starting from the Yule-Simon distribution. A modeling is proposed leading to a universal form. A theoretical suggestion for the “best (or optimal) distribution”, is provided through an entropy argument. The ranking of areas through the number of cities in various countries and some sport competition ranking serves for the present illustrations

Scaling and Universality in City Space Syntax: between Zipf and Matthew

We report about universality of rank-integration distributions of open spaces in city space syntax similar to the famous rank-size distributions of cities (Zipf's law). We also demonstrate that the degree of choice an open space represents for other spaces directly linked to it in a city follows a power law statistic. Universal statistical behavior of space syntax measures uncovers the universality of the city creation mechanism. We suggest that the observed universality may help to establish the international definition of a city as a specific land use pattern.Comment: 24 pages, 5 *.eps figure

Universal scaling of the magnetic anisotropy in two-dimensional rare-earth layers

Unraveling the influence that low dimensionality has upon the spin's stability in two-dimensional (2D) systems is instrumental for the efficient engineering of energy barriers in ultrathin magnetic layers. Taking rare-earth-based ultrathin multilayered nanostructures as a model system, we have investigated the dissimilar impact that low dimensionality and finite-size effects have upon the magnetic anisotropy energy (MAE) at the nanoscale. We conclusively show that the reduced dimensionality of the spin's system in 2D ferromagnetic layers imprints on the MAE constants a universal temperature decay as a quadratic power law of the reduced magnetization. This result is in agreement with predictions, although in marked contrast to the rank-dependent, thereby faster, decay of the MAE constants observed in three-dimensional nanostructures

Scaling of Geographic Space as a Universal Rule for Map Generalization

Map generalization is a process of producing maps at different levels of detail by retaining essential properties of the underlying geographic space. In this paper, we explore how the map generalization process can be guided by the underlying scaling of geographic space. The scaling of geographic space refers to the fact that in a geographic space small things are far more common than large ones. In the corresponding rank-size distribution, this scaling property is characterized by a heavy tailed distribution such as a power law, lognormal, or exponential function. In essence, any heavy tailed distribution consists of the head of the distribution (with a low percentage of vital or large things) and the tail of the distribution (with a high percentage of trivial or small things). Importantly, the low and high percentages constitute an imbalanced contrast, e.g., 20 versus 80. We suggest that map generalization is to retain the objects in the head and to eliminate or aggregate those in the tail. We applied this selection rule or principle to three generalization experiments, and found that the scaling of geographic space indeed underlies map generalization. We further relate the universal rule to T\"opfer's radical law (or trained cartographers' decision making in general), and illustrate several advantages of the universal rule. Keywords: Head/tail division rule, head/tail breaks, heavy tailed distributions, power law, and principles of selectionComment: 12 pages, 9 figures, 4 table

Earthquakes economic costs through rank-size laws

© 2017 IOP Publishing Ltd and SISSA Medialab srl. This paper is devoted to assessing the presence of some regularities in the magnitudes of the earthquakes in Italy between January 24th, 2016 and January 24th, 2017, and to propose an earthquakes cost indicator. The considered data includes the catastrophic events in Amatrice and in the Marche region. To our purpose, we implement two typologies of rank-size analysis: the classical Zipf-Mandelbrot law and the so-called universal law proposed by Ausloos and Cerqueti (2016 PLoS One 11 e0166011). The proposed generic measure of the economic impact of earthquakes moves from the assumption of the existence of a cause-effect relation between earthquakes magnitudes and economic costs. At this aim, we hypothesize that such a relation can be formalized in a functional way to show how infrastructure resistance affects the cost. Results allow us to clarify the impact of an earthquake on the social context and might serve to strengthen the struggle against the dramatic outcomes of such natural phenomena

Zipf's law, 1/f noise, and fractal hierarchy

Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic events are typical ubiquitous general empirical observations across the individual sciences which cannot be understood within the set of references developed within the specific scientific domains. All these observations are associated with scaling laws and have caused a broad research interest in the scientific circle. However, the inherent relationships between these scaling phenomena are still pending questions remaining to be researched. In this paper, theoretical derivation and mathematical experiments are employed to reveal the analogy between fractal patterns, 1/f noise, and the Zipf distribution. First, the multifractal process follows the generalized Zipf's law empirically. Second, a 1/f spectrum is identical in mathematical form to Zipf's law. Third, both 1/f spectra and Zipf's law can be converted into a self-similar hierarchy. Fourth, fractals, 1/f spectra, Zipf's law, and the occurrence of large catastrophic events can be described with similar exponential laws and power laws. The self-similar hierarchy is a more general framework or structure which can be used to encompass or unify different scaling phenomena and rules in both physical and social systems such as cities, rivers, earthquakes, fractals, 1/f noise, and rank-size distributions. The mathematical laws on the hierarchical structure can provide us with a holistic perspective of looking at complexity such as self-organized criticality (SOC).Comment: 20 pages, 9 figures, 3 table

Universal scaling in sports ranking

Ranking is a ubiquitous phenomenon in the human society. By clicking the web pages of Forbes, you may find all kinds of rankings, such as world's most powerful people, world's richest people, top-paid tennis stars, and so on and so forth. Herewith, we study a specific kind, sports ranking systems in which players' scores and prize money are calculated based on their performances in attending various tournaments. A typical example is tennis. It is found that the distributions of both scores and prize money follow universal power laws, with exponents nearly identical for most sports fields. In order to understand the origin of this universal scaling we focus on the tennis ranking systems. By checking the data we find that, for any pair of players, the probability that the higher-ranked player will top the lower-ranked opponent is proportional to the rank difference between the pair. Such a dependence can be well fitted to a sigmoidal function. By using this feature, we propose a simple toy model which can simulate the competition of players in different tournaments. The simulations yield results consistent with the empirical findings. Extensive studies indicate the model is robust with respect to the modifications of the minor parts.Comment: 8 pages, 7 figure