Stretched exponential distributions in nature and economy: ``fat tails'' with characteristic scales
TL;DR: In this paper, the authors proposed the stretched exponential family as a complement to the often used power law distributions, which has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation.
Abstract: To account quantitatively for many reported “natural” fat tail distributions in Nature and Economy, we propose the stretched exponential family as a complement to the often used power law distributions. It has many advantages, among which to be economical with only two adjustable parameters with clear physical interpretation. Furthermore, it derives from a simple and generic mechanism in terms of multiplicative processes. We show that stretched exponentials describe very well the distributions of radio and light emissions from galaxies, of US GOM OCS oilfield reserve sizes, of World, US and French agglomeration sizes, of country population sizes, of daily Forex US-Mark and Franc-Mark price variations, of Vostok (near the south pole) temperature variations over the last 400 000 years, of the Raup-Sepkoski's kill curve and of citations of the most cited physicists in the world. We also discuss its potential for the distribution of earthquake sizes and fault displacements. We suggest physical interpretations of the parameters and provide a short toolkit of the statistical properties of the stretched exponentials. We also provide a comparison with other distributions, such as the shifted linear fractal, the log-normal and the recently introduced parabolic fractal distributions.
Citations
More filters
TL;DR: The index h, defined as the number of papers with citation number ≥h, is proposed as a useful index to characterize the scientific output of a researcher.
Abstract: I propose the index h, defined as the number of papers with citation number ≥h, as a useful index to characterize the scientific output of a researcher.
8,996 citations
TL;DR: The recent rapid progress in the statistical physics of evolving networks is reviewed, and how growing networks self-organize into scale-free structures is discussed, and the role of the mechanism of preferential linking is investigated.
Abstract: We review the recent rapid progress in the statistical physics of evolving networks. Interest has focused mainly on the structural properties of complex networks in communications, biology, social sciences and economics. A number of giant artificial networks of this kind have recently been created, which opens a wide field for the study of their topology, evolution, and the complex processes which occur in them. Such networks possess a rich set of scaling properties. A number of them are scale-free and show striking resilience against random breakdowns. In spite of the large sizes of these networks, the distances between most of their vertices are short - a feature known as the 'small-world' effect. We discuss how growing networks self-organize into scale-free structures, and investigate the role of the mechanism of preferential linking. We consider the topological and structural properties of evolving networks, and percolation and disease spread on these networks. We present a number of models demonstrat...
3,368 citations
Cites methods from "Stretched exponential distributions..."
...dicated from similar estimation that these data are also consistent with the k−y i exp[−constk 1−y i ] form of the distribution if one sets y= 0.9 for the ISI net and y= 0.7 for Phys. Rev. D. In Ref. [26], the very tail of a different distribution was studied. The ranking dependence of the number of citations to the 1120most cited physicists was described by a 6 stretched exponential function. Of cours...
[...]
TL;DR: The model of growing networks with the preferential attachment of new links is generalized to include initial attractiveness of sites and it is shown that the relation beta(gamma-1) = 1 between the exponents is universal.
Abstract: The model of growing networks with the preferential attachment of new links is generalized to include initial attractiveness of sites. We find the exact form of the stationary distribution of the number of incoming links of sites in the limit of long times, $P(q)$, and the long-time limit of the average connectivity $\overline{q}(s,t)$ of a site $s$ at time $t$ (one site is added per unit of time). At long times, $P(q)\ensuremath{\sim}{q}^{\ensuremath{-}\ensuremath{\gamma}}$ at $q\ensuremath{\rightarrow}\ensuremath{\infty}$ and $\overline{q}(s,t)\ensuremath{\sim}(s/t{)}^{\ensuremath{-}\ensuremath{\beta}}$ at $s/t\ensuremath{\rightarrow}0$, where the exponent $\ensuremath{\gamma}$ varies from $2$ to $\ensuremath{\infty}$ depending on the initial attractiveness of sites. We show that the relation $\ensuremath{\beta}(\ensuremath{\gamma}\ensuremath{-}1)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1$ between the exponents is universal.
1,236 citations
TL;DR: In this paper, the authors provide guidelines for the accurate and practical estimation of exponents and fractal dimensions of natural fracture systems, including length, displacement and aperture power law exponents.
Abstract: Scaling in fracture systems has become an active field of research in the last 25 years motivated by practical applications in hazardous waste disposal, hy- drocarbon reservoir management, and earthquake haz- ard assessment. Relevant publications are therefore spread widely through the literature. Although it is rec- ognized that some fracture systems are best described by scale-limited laws (lognormal, exponential), it is now recognized that power laws and fractal geometry provide widely applicable descriptive tools for fracture system characterization. A key argument for power law and fractal scaling is the absence of characteristic length scales in the fracture growth process. All power law and fractal characteristics in nature must have upper and lower bounds. This topic has been largely neglected, but recent studies emphasize the importance of layering on all scales in limiting the scaling characteristics of natural fracture systems. The determination of power law expo- nents and fractal dimensions from observations, al- though outwardly simple, is problematic, and uncritical use of analysis techniques has resulted in inaccurate and even meaningless exponents. We review these tech- niques and suggest guidelines for the accurate and ob- jective estimation of exponents and fractal dimensions. Syntheses of length, displacement, aperture power law exponents, and fractal dimensions are found, after crit- ical appraisal of published studies, to show a wide vari- ation, frequently spanning the theoretically possible range. Extrapolations from one dimension to two and from two dimensions to three are found to be nontrivial, and simple laws must be used with caution. Directions for future research include improved techniques for gathering data sets over great scale ranges and more rigorous application of existing analysis methods. More data are needed on joints and veins to illuminate the differences between different fracture modes. The phys- ical causes of power law scaling and variation in expo- nents and fractal dimensions are still poorly understood.
1,153 citations
Cites background from "Stretched exponential distributions..."
...An alternative to the power and exponential laws is the stretched exponential that plays an intermediate role [Laherrere and Sornette, 1998]....
[...]
...49 LaPointe [1988] joints 3 modified box-counting D 5 2....
[...]
Book•
01 Jan 2006
TL;DR: Jones and Baumgartner as discussed by the authors studied how politicians manage the flood of information from a wide range of sources, and which issues do they pay attention to and why, in American politics.
Abstract: On any given day, policymakers are required to address a multitude of problems and make decisions about a variety of issues, from the economy and education to health care and defense. This has been true for years, but until now no studies have been conducted on how politicians manage the flood of information from a wide range of sources. How do they interpret and respond to such inundation? Which issues do they pay attention to and why? Bryan D. Jones and Frank R. Baumgartner answer these questions on decision-making processes and prioritization in "The Politics of Attention". Analyzing fifty years of data, Jones and Baumgartner's book is the first study of American politics based on a new information-processing perspective. The authors bring together the allocation of attention and the operation of governing institutions into a single model that traces public policies, public and media attention to them, and governmental decisions across multiple institutions. "The Politics of Attention offers a groundbreaking approach to American politics based on the responses of policymakers to the flow of information. It asks how the system solves, or fails to solve, problems rather than looking to how individual preferences are realized through political action.
884 citations
References
More filters
Book•
01 Jan 1982
TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
Abstract: "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature
24,199 citations
"Stretched exponential distributions..." refers background in this paper
...On the other hand, the relevance of power laws in Nature is less clear-cut even if it has repeatedly been claimed to describe many natural phenomena [1, 6 ,7]....
[...]
Book•
01 Jan 1982
TL;DR: In this article, exactly solved models of statistical mechanics are discussed. But they do not consider exactly solvable models in statistical mechanics, which is a special issue in the statistical mechanics of the classical two-dimensional faculty of science.
Abstract: exactly solved models in statistical mechanics exactly solved models in statistical mechanics rodney j baxter exactly solved models in statistical mechanics exactly solved models in statistical mechanics flae exactly solved models in statistical mechanics dover books exactly solved models in statistical mechanics dover books exactly solved models in statistical mechanics dover books hatsutori in size 15 gvg7bzbookyo.qhigh literature cited r. j. baxter, exactly solved models in exactly solvable models in statistical mechanics exactly solved models in statistical mechanics dover books okazaki in size 24 vk19j3book.buncivy exactly solved models of statistical mechanics valerio nishizawa in size 11 b4zntdbookntey fukuda in size 13 33oloxbooknhuy yamada in size 19 x6g84ybook.zolay in honour of r j baxter’s 75th birthday arxiv:1608.04899v2 statistical mechanics, threedimensionality and np beautiful models: 70 years of exactly solved quantum many exactly solved models in statistical mechanics (dover solved lattice models: 1944 2010 university of melbourne exactly solved models and beyond: a special issue in the statistical mechanics of the classical two-dimensional faculty of science, p. j. saf ́arik university in ko?sice? a one-dimensional statistical mechanics model with exact statistical mechanics department of physics and astronomy statistical mechanics principles and selected applications graph theory and statistical physics yaroslavvb chapter 4 methods of statistical mechanics ijs thermodynamics and an introduction to thermostatistics potts models and related problems in statistical mechanics methods of quantum field theory in statistical physics statistical mechanics: theory and molecular simulation exactly solvable su(n) mixed spin ladders springer statistical field theory : an introduction to exactly
7,761 citations
"Stretched exponential distributions..." refers background in this paper
...[ 2 ], with an abondance of numerical evidence for instance for the distribution of percolation clusters at criticality [3] and for many other models in statistical physics....
[...]
Book•
01 Jan 1992
TL;DR: In this article, a scaling solution for the Bethe lattice is proposed for cluster numbers and a scaling assumption for cluster number scaling assumptions for cluster radius and fractal dimension is proposed.
Abstract: Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques
7,349 citations
Book•
01 Jan 19495,898 citations
Book•
01 Jan 1996
TL;DR: This chapter discusses the Sandpile Paradigm, Earthquakes, Starquakes, and Solar Flares, and the "Game of Life": Complexity Is Criticality, and Is Life a Self-Organized Critical Phenomenon?
Abstract: 1: Complexity and criticality. 2: The discovery of self-organized criticality. 3: The sandpile paradigm. 4: Real sandpiles and landscape formation. 5: Earthquakes, starquakes, and solar flares. 6: The "Game of Life": complexity is critical. 7: Is life a self-organized critical phenomenon. 8: Mass extinctions and punctuated equilibria in a simple model of evolution. 9: Theory of the punctuated equilibrium model. 10: The brain. 11: On economies and traffic jams. Bibliography. Index
2,465 citations