Recently, I became interested in a current debate over whether file size distributions are best modelled by a power law distribution or a a lognormal distribution. In trying

Abstract We give a plausible explanation of the power law distributions of degrees observed in the graphs arising in the Internet topology [5] based on a toy model of Internet growth in which two objectives are optimized simultaneously: "last mile " connection costs, and transmission delays measured in hops. We also point out a similar phenomenon, anticipated in [2], in the distribution of file sizes. Our results seem to suggest that power laws tend to arise as a result of complex, multi-objective optimization.

Abstract. In this paper, we introduce and analyze a new, dynamic generative user model to explain the behavior of file size distributions. Our Recursive Forest File model combines multiplicative models that generate lognormal distributions with recent work on random graph models for the web. Unlike similar previous work, our Recursive Forest File model allows new files to be created and old files to be deleted over time, and our analysis covers problematic issues such as correlation among file sizes. Moreover, our model allows natural variations where files that are copied or modified are more likely to be copied or modified subsequently. Previous empirical work suggests that file sizes tend to have a lognormal body but a Pareto tail. The Recursive Forest File model explains this behavior, yielding a double Pareto distribution, which has a Pareto tail but close to a lognormal body. We believe the Recursive Forest model may be useful for describing other power law phenomena in computer systems as well as other fields. 1.

We present a broad, phenomenological picture of the distribution of the length of urban linear segments, l, derived from maps of 36 cities in 14 different countries. By scaling the Zipf plot of l, we obtain two master curves for a sample of cities, which are not a function of city size. We show that a third class of cities is not easily classifiable into these two universality classes. The cumulative distribution of l displays power-law tails with two distinct exponents, and. We suggest a link between our observations and the possibility of observing and modelling urban growth using Levy processes. 1.

We study the dynamics of the distribution of wealth in an overlapping generation economy with finitely lived agents and inter-generational transmission of wealth. Financial markets are incomplete, exposing agents to both labor and capital income risk. We show that the stationary wealth distribution is a Pareto distribution in the right tail and that it is capital income risk, rather than labor income, that drives the properties of the right tail of the wealth distribution. We also study analytically the dependence of the distribution of wealth, of wealth inequality in particular, on various fiscal policy instruments like capital income taxes and estate taxes, and on different degrees of social mobility. We show that capital income and estate taxes can significantly reduce wealth inequality, as do institutions favoring social mobility. Finally, we calibrate the economy to match the Lorenz curve of the wealth distribution of the U.S. economy.

informs ® doi 10.1287/moor.1030.0071 © 2004 INFORMS Since growth stocks tend to have low or even negative earnings and high volatility, it is a great challenge to derive a meaningful mathematical model within the traditional valuation framework. This paper attempts to derive a suitable diffusion model for growth stocks by using the idea of size distribution. Numerical illustration of the model based on the data covering the time period of the recent boom and burst of the “Internet bubble ” is also presented. Key words: Feller process; CIR model; power law; steady-state distribution; weak convergence; decay rate; regression

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A large consensus now seems to take for granted that the distributions of empirical returns of financial time series are regularly varying, with a tail exponent b close to 3. First, we show by synthetic tests performed on time series with time dependence in the volatility with both Pareto and Stretched-Exponential distributions that for sample of moderate size, the standard generalized extreme value (GEV) estimator is quite inefficient due to the possibly slow convergence toward the asymptotic theoretical distribution and the existence of biases in presence of dependence between data. Thus it cannot distinguish reliably between rapidly and regularly varying classes of distributions. The Generalized Pareto distribution (GPD) estimator works better, but still lacks power in the presence of strong dependence. Then, we use a parametric representation of the tail of the distributions of returns of 100 years of daily return of the Dow Jones Industrial Average and over 1 years of 5-minutes returns of the Nasdaq Composite index, encompassing both a regularly varying distribution in one limit of the parameters and rapidly varying distributions of the class of the Stretched-Exponential (SE) and Log-Weibull distributions in other limits. Using the method of nested hypothesis testing (Wilks ’ theorem),

The emerging sub#eld of econophysics explores the degree to which certain concepts and methods from statistical physics can be appropriately modi#ed and adapted to provide new insights into questions that have been the focus of interest in the economics community. Here we give a brief overview of two examples of research topics that are receiving recent attention. A #rst topic is the characterization of the dynamics of stock price #uctuations. For example, we investigate the relation between trading activity -- measured by the number of transactions N#t -- and the price change G#t for a given stock, over a time interval [t; t +#t]. We relate the time-dependent standard deviation of price #uctuations -- volatility -- to two microscopic quantities: the number of transactions N#t in #t and the variance W #t of the price changes for all transactions in #t. Our work indicates that while the pronounced tails in the distribution of price #uctuations arise from W#t , the long-range correlations found in |G#t | are largely due to N#t .We also investigate the relation between price #uctuations and the number of shares Q#t traded in #t. We #nd that the distribution of Q#t is consistent with a stable L#evy distribution, suggesting aL#evy scaling relationship between Q#t and N#t , which would provide one explanation for volume-volatility co-movement. A second topic concerns cross-correlations between the price #uctuations of di#erent stocks. We adapt a conceptual framework, random matrix theory (RMT), #rst used in physics to interpret statistical properties of nuclear energy spectra. RMT makes predictions for the statistical properties of matrices that are universal, that is, do not depend on the interactions between the elements comprising the system. In physics systems, deviat...

The detachment of cells from bacterial biofilms is an important, yet poorly understood and largely unquantified phenomenon. Detached cell clumps from medical devices may form microemboli and lead to metastasis, especially if they are resistant to host defenses and antibiotics. In manufacturing plants detached clumps entering a process stream decrease product quality. Two strains of Pseudomonas aeruginosa, a wild type (PAO1) and a cell signaling mutant (JP1), were studied to (i) quantify and model detachment patterns and (ii) determine the influence of cell signaling on detachment. We collected effluent from a biofilm flowthrough reactor and determined the size distribution for cell detachment events by microscopic examination and image analysis. The two strains were similar in terms of both biofilm structure and detachment patterns. Most of the detachment events were single-cell events; however, multiple-cell detachment events contributed a large fraction of the total detached cells. The rates at which events containing multiple cells detached from the biofilm were estimated by fitting a statistical model to the size distribution data. For events consisting of at least 1,000 cells, the estimated rates were 4.5 events mm �2 min �1 for PAO1 and 4.3 events mm �2 min �1 for JP1. These rates may be significant when they are scaled up to the total area of a real biofilm-contaminated medical device surface and to the hours or days of patient exposure. The detachment of cells from a bacterial biofilm attached to