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

The Mathematical Relationship between Zipf's Law and the Hierarchical Scaling Law

The empirical studies of city-size distribution show that Zipf's law and the hierarchical scaling law are linked in many ways. The rank-size scaling and hierarchical scaling seem to be two different sides of the same coin, but their relationship has never been revealed by strict mathematical proof. In this paper, the Zipf's distribution of cities is abstracted as a q-sequence. Based on this sequence, a self-similar hierarchy consisting of many levels is defined and the numbers of cities in different levels form a geometric sequence. An exponential distribution of the average size of cities is derived from the hierarchy. Thus we have two exponential functions, from which follows a hierarchical scaling equation. The results can be statistically verified by simple mathematical experiments and observational data of cities. A theoretical foundation is then laid for the conversion from Zipf's law to the hierarchical scaling law, and the latter can show more information about city development than the former. Moreover, the self-similar hierarchy provides a new perspective for studying networks of cities as complex systems. A series of mathematical rules applied to cities such as the allometric growth law, the 2^n principle and Pareto's law can be associated with one another by the hierarchical organization.Comment: 30 pages, 5 figures, 5 tables, Physica A: Statistical Mechanics and its Applications, 201

The Accelerating Growth of Online Tagging Systems

Research on the growth of online tagging systems not only is interesting in its own right, but also yields insights for website management and semantic web analysis. Traditional models that describing the growth of online systems can be divided between linear and nonlinear versions. Linear models, including the BA model (Brabasi and Albert, 1999), assume that the average activity of users is a constant independent of population. Hence the total activity is a linear function of population. On the contrary, nonlinear models suggest that the average activity is affected by the size of the population and the total activity is a nonlinear function of population. In the current study, supporting evidences for the nonlinear growth assumption are obtained from data on Internet users' tagging behavior. A power law relationship between the number of new tags (F) and the population (P), which can be expressed as F ~ P ^ gamma (gamma > 1), is found. I call this pattern accelerating growth and find it relates the to time-invariant heterogeneity in individual activities. I also show how a greater heterogeneity leads to a faster growth.Comment: 8 pages, 3 figure

Languages cool as they expand: Allometric scaling and the decreasing need for new words

We analyze the occurrence frequencies of over 15 million words recorded in millions of books published during the past two centuries in seven different languages. For all languages and chronological subsets of the data we confirm that two scaling regimes characterize the word frequency distributions, with only the more common words obeying the classic Zipf law. Using corpora of unprecedented size, we test the allometric scaling relation between the corpus size and the vocabulary size of growing languages to demonstrate a decreasing marginal need for new words, a feature that is likely related to the underlying correlations between words. We calculate the annual growth fluctuations of word use which has a decreasing trend as the corpus size increases, indicating a slowdown in linguistic evolution following language expansion. This ‘‘cooling pattern’’ forms the basis of a third statistical regularity, which unlike the Zipf and the Heaps law, is dynamical in nature

A triple helix model of medical innovation: supply, demand, and technological capabilities in terms of medical subject headings

We develop a model of innovation that enables us to trace the interplay among three key dimensions of the innovation process: (i) demand of and (ii) supply for innovation, and (iii) technological capabilities available to generate innovation in the forms of products, processes, and services. Building on triple helix research, we use entropy statistics to elaborate an indicator of mutual information among these dimensions that can provide indication of reduction of uncertainty. To do so, we focus on the medical context, where uncertainty poses significant challenges to the governance of innovation. We use the Medical Subject Headings (MeSH) of MEDLINE/PubMed to identify publications classified within the categories “Diseases" (C), "Drugs and Chemicals" (D), "Analytic, Diagnostic, and Therapeutic Techniques and Equipment" (E) and use these as knowledge representations of demand, supply, and technological capabilities, respectively. Three case-studies of medical research areas are used as representative 'entry perspectives' of the medical innovation process. These are: (i) human papilloma virus, (ii) RNA interference, and (iii) magnetic resonance imaging. We find statistically significant periods of synergy among demand, supply, and technological capabilities (C-D-E) that point to three-dimensional interactions as a fundamental perspective for the understanding and governance of the uncertainty associated with medical innovation. Among the pairwise configurations in these contexts, the demand-technological capabilities (C-E) provided the strongest link, followed by the supply-demand (D-C) and the supply-technological capabilities (D-E) channels

Structure and non-structure of centrosomal proteins

Here we perform a large-scale study of the structural properties and the expression of proteins that constitute the human Centrosome. Centrosomal proteins tend to be larger than generic human proteins (control set), since their genes contain in average more exons (20.3 versus 14.6). They are rich in predicted disordered regions, which cover 57% of their length, compared to 39% in the general human proteome. They also contain several regions that are dually predicted to be disordered and coiled-coil at the same time: 55 proteins (15%) contain disordered and coiled-coil fragments that cover more than 20% of their length. Helices prevail over strands in regions homologous to known structures (47% predicted helical residues against 17% predicted as strands), and even more in the whole centrosomal proteome (52% against 7%), while for control human proteins 34.5% of the residues are predicted as helical and 12.8% are predicted as strands. This difference is mainly due to residues predicted as disordered and helical (30% in centrosomal and 9.4% in control proteins), which may correspond to alpha-helix forming molecular recognition features (α-MoRFs). We performed expression assays for 120 full-length centrosomal proteins and 72 domain constructs that we have predicted to be globular. These full-length proteins are often insoluble: Only 39 out of 120 expressed proteins (32%) and 19 out of 72 domains (26%) were soluble. We built or retrieved structural models for 277 out of 361 human proteins whose centrosomal localization has been experimentally verified. We could not find any suitable structural template with more than 20% sequence identity for 84 centrosomal proteins (23%), for which around 74% of the residues are predicted to be disordered or coiled-coils. The three-dimensional models that we built are available at http://ub.cbm.uam.es/centrosome/models/index.php

Evolution of scaling emergence in large-scale spatial epidemic spreading

Background: Zipf's law and Heaps' law are two representatives of the scaling concepts, which play a significant role in the study of complexity science. The coexistence of the Zipf's law and the Heaps' law motivates different understandings on the dependence between these two scalings, which is still hardly been clarified. Methodology/Principal Findings: In this article, we observe an evolution process of the scalings: the Zipf's law and the Heaps' law are naturally shaped to coexist at the initial time, while the crossover comes with the emergence of their inconsistency at the larger time before reaching a stable state, where the Heaps' law still exists with the disappearance of strict Zipf's law. Such findings are illustrated with a scenario of large-scale spatial epidemic spreading, and the empirical results of pandemic disease support a universal analysis of the relation between the two laws regardless of the biological details of disease. Employing the United States(U.S.) domestic air transportation and demographic data to construct a metapopulation model for simulating the pandemic spread at the U.S. country level, we uncover that the broad heterogeneity of the infrastructure plays a key role in the evolution of scaling emergence. Conclusions/Significance: The analyses of large-scale spatial epidemic spreading help understand the temporal evolution of scalings, indicating the coexistence of the Zipf's law and the Heaps' law depends on the collective dynamics of epidemic processes, and the heterogeneity of epidemic spread indicates the significance of performing targeted containment strategies at the early time of a pandemic disease.Comment: 24pages, 7figures, accepted by PLoS ON

Síndrome de Budd Chiari e Gravidez

Introdução: Na literatura são raros os casos publicados sobre a evolução de uma gestação com o diagnóstico de Síndrome de Budd Chiari e hipertensão portal conhecida antes da gravidez, assim como de sequelas fetais inerentes à terapêutica associada a esta patologia. Caso Clínico: Grávida com o diagnóstico de Síndrome de Budd chiari associado a uma gastropatia hipertensiva portal grave, a um défice de proteína C e factor V Leiden. Manteve terapêutica durante toda a gravidez com hidroxiureia e propranolol. Não se verificaram quaisquer intercorrências na evolução da gestação. A hemorragia persistente da cicatriz operatória condicionou a suspensão da anticoagulação facilitando a retrombose das veias supra-hepáticas e consequente agravamento do quadro clínico. Comentários: É possível conseguir-se um recém-nascido saudável mantendo-se a mãe clinicamente estável. Uma anticoagulação eficaz é um factor decisivo na prevenção do acidente trombótico

Episiotomy. Routine Versus Selective Use

A episiotomia é um procedimento cirúrgico quase universal que foi introduzido na prática clínica sem evidência científica que suportasse o seu benefício. O seu uso continua a ser rotineiro apesar de não cumprir a maioria dos objectivos pelos quais é justificado, isto é, não diminui o risco de lesões perineais severas, não previne o desenvolvimento de relaxamento pélvico e não tem impacto sobre a morbilidade ou mortalidade do recém nascido. Múltiplos estudos, retrospectivos e prospectivos, têm comparado o uso generalizado versus selectivo da episiotomia e, ao contrário do defendido por vários autores e durante várias décadas, o uso selectivo está associado a melhores resultados, como por exemplo, uma diminuição das lesões severas e um maior número de períneos intactos, embora se tenha também verificado um aumento do número de lacerações anteriores. Assim, parece ser correcto recomendar um uso selectivo da episiotomia, sendo 30% o valor sugerido por alguns autores, tornando-se, então, importante apostar em novas técnicas não cirúrgicas que ajudem a obter uma integridade perineal. Mais estudos são imprescindíveis e eticamente necessários para esclarecerem quais são, de facto, as verdadeiras indicações para a realização deste acto. Um uso ponderado, com decisões caso a caso, parece ser a atitude mais correcta enquanto aguardamos por uma maior evidência científica