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"Power laws"
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FROM BLACK POWER TO PRISON POWER : the making of Jones v. North Carolina Prisoners' Labor Union
\"This book uses the landmark case Jones v. North Carolina Prisoners' Labor Union to examine the strategies of prison inmates using race and radicalism to inspire the formation of an inmate labor union. It thus rekindles the debate over the triumphs and troubles associated with the use of Black Power as a platform for influencing legal policy and effecting change for inmates. While the ideology of the prison rights movement was complex, it rested on the underlying principle that the right to organize, and engage in political dissidence, was not only a First Amendment right guaranteed to free blacks, but one that should be explicitly guaranteed to captive blacks--a point too often overlooked in previous analyses. Ultimately, this seminal case study not only illuminates the history of Black Power but that of the broader prisoners' rights movement as well\"-- Provided by publisher.
Book
True scale-free networks hidden by finite size effects
We analyze about 200 naturally occurring networks with distinct dynamical origins to formally test whether the commonly assumed hypothesis of an underlying scale-free structure is generally viable. This has recently been questioned on the basis of statistical testing of the validity of power law distributions of network degrees. Specifically, we analyze by finite size scaling analysis the datasets of real networks to check whether the purported departures from power law behavior are due to the finiteness of sample size. We find that a large number of the networks follows a finite size scaling hypothesis without any self-tuning. This is the case of biological protein interaction networks, technological computer and hyperlink networks, and informational networks in general. Marked deviations appear in other cases, especially involving infrastructure and transportation but also in social networks. We conclude that underlying scale invariance properties of many naturally occurring networks are extant features often clouded by finite size effects due to the nature of the sample data.
Journal Article
Power Laws in Economics: An Introduction
Many of the insights of economics seem to be qualitative, with many fewer reliable quantitative laws. However a series of power laws in economics do count as true and nontrivial quantitative laws—and they are not only established empirically, but also understood theoretically. I will start by providing several illustrations of empirical power laws having to do with patterns involving cities, firms, and the stock market. I summarize some of the theoretical explanations that have been proposed. I suggest that power laws help us explain many economic phenomena, including aggregate economic fluctuations. I hope to clarify why power laws are so special, and to demonstrate their utility. In conclusion, I list some power-law-related economic enigmas that demand further exploration. A formal definition may be useful.
Journal Article
Predator–prey power laws: trophic interactions give rise to scale-invariant ecosystems
Scaling laws and power-law distributions are ubiquitous in ecological systems. However, it is not clear what factors give rise to such universal regularities. Here, I show scaling laws are a simple consequence of scale-invariant distributions, and both result from simple commonalities of diverse ecosystems. I introduce a simple model of predator–prey interactions in which predators and prey move on a two-dimensional space in search of resources that they use to survive and reproduce. As primary resources increase, the food web exhibits a series of transitions to phases with equilibrium dynamics and top-down control of the food web, non-equilibrium dynamics with bottom-up control, and unstable dynamics exhibiting the paradox of enrichment. The model shows resource heterogeneity can solve the paradox of enrichment and ensure the stability of ecosystems. Scale-invariant spatial distribution of prey and predators and a surprisingly rich set of scaling laws, including predator–prey and Taylor’s power laws, appear in the non-equilibrium phase. The model predicts both Taylor’s power law and predator–prey power law can be extended to a rich set of fluctuation scaling laws governing the fluctuation of predator’s and prey’s densities and growth. A mathematical theory suggests scaling laws result from the scale-invariance of the spatial distribution of prey and predators.
Journal Article
Power-law distribution of degree–degree distance
Whether real-world complex networks are scale free or not has long been controversial. Recently, in Broido and Clauset [A. D. Broido, A. Clauset, Nat. Commun. 10, 1017 (2019)], it was claimed that the degree distributions of real-world networks are rarely power law under statistical tests. Here, we attempt to address this issue by defining a fundamental property possessed by each link, the degree–degree distance, the distribution of which also shows signs of being power law by our empirical study. Surprisingly, although full-range statistical tests show that degree distributions are not often power law in real-world networks, we find that in more than half of the cases the degree–degree distance distributions can still be described by power laws. To explain these findings, we introduce a bidirectional preferential selection model where the link configuration is a randomly weighted, two-way selection process. The model does not always produce solid power-law distributions but predicts that the degree–degree distance distribution exhibits stronger power-law behavior than the degree distribution of a finite-size network, especially when the network is dense. We test the strength of our model and its predictive power by examining how real-world networks evolve into an overly dense stage and how the corresponding distributions change. We propose that being scale free is a property of a complex network that should be determined by its underlying mechanism (e.g., preferential attachment) rather than by apparent distribution statistics of finite size. We thus conclude that the degree–degree distance distribution better represents the scale-free property of a complex network.
Journal Article
POWER-LAW DISTRIBUTIONS IN BINNED EMPIRICAL DATA
Many man-made and natural phenomena, including the intensity of earthquakes, population of cities and size of international wars, are believed to follow power-law distributions. The accurate identification of power-law patterns has significant consequences for correctly understanding and modeling complex systems. However, statistical evidence for or against the power-law hypothesis is complicated by large fluctuations in the empirical distribution's tail, and these are worsened when information is lost from binning the data. We adapt the statistically principled framework for testing the power-law hypothesis, developed by Clauset, Shalizi and Newman, to the case of binned data. This approach includes maximum-likelihood fitting, a hypothesis test based on the Kolmogorov–Smirnov goodness-of-fit statistic and likelihood ratio tests for comparing against alternative explanations. We evaluate the effectiveness of these methods on synthetic binned data with known structure, quantify the loss of statistical power due to binning, and apply the methods to twelve real-world binned data sets with heavy-tailed patterns.
Journal Article
Scale‐free dynamics of core‐periphery topography
The human brain's cerebral cortex exhibits a topographic division into higher‐order transmodal core and lower‐order unimodal periphery regions. While timescales between the core and periphery region diverge, features of their power spectra, especially scale‐free dynamics during resting‐state and their mdulation in task states, remain unclear. To answer this question, we investigated the ~1/f‐like pink noise manifestation of scale‐free dynamics in the core‐periphery topography during rest and task states applying infra‐slow inter‐trial intervals up to 1 min falling inside the BOLD's infra‐slow frequency band. The results demonstrate (1) higher resting‐state power‐law exponent (PLE) in the core compared to the periphery region; (2) significant PLE increases in task across the core and periphery regions; and (3) task‐related PLE increases likely followed the task's atypically low event rates, namely the task's periodicity (inter‐trial interval = 52–60 s; 0.016–0.019 Hz). A computational model and a replication dataset that used similar infra‐slow inter‐trial intervals provide further support for our main findings. Altogether, the results show that scale‐free dynamics differentiate core and periphery regions in the resting‐state and mediate task‐related effects. Scale‐free dynamics are investigated in the cerebral cortex's core‐periphery division in fMRI. Both rest and task states were assessed. We demonstrate that the brain's scale‐free dynamics are modulated by the task's periodicity in the infra‐slow band.
Journal Article
Crossover from anomalous to normal diffusion: truncated power-law noise correlations and applications to dynamics in lipid bilayers
The emerging diffusive dynamics in many complex systems show a characteristic crossover behaviour from anomalous to normal diffusion which is otherwise fitted by two independent power-laws. A prominent example for a subdiffusive-diffusive crossover are viscoelastic systems such as lipid bilayer membranes, while superdiffusive-diffusive crossovers occur in systems of actively moving biological cells. We here consider the general dynamics of a stochastic particle driven by so-called tempered fractional Gaussian noise, that is noise with Gaussian amplitude and power-law correlations, which are cut off at some mesoscopic time scale. Concretely we consider such noise with built-in exponential or power-law tempering, driving an overdamped Langevin equation (fractional Brownian motion) and fractional Langevin equation motion. We derive explicit expressions for the mean squared displacement and correlation functions, including different shapes of the crossover behaviour depending on the concrete tempering, and discuss the physical meaning of the tempering. In the case of power-law tempering we also find a crossover behaviour from faster to slower superdiffusion and slower to faster subdiffusion. As a direct application of our model we demonstrate that the obtained dynamics quantitatively describes the subdiffusion-diffusion and subdiffusion-subdiffusion crossover in lipid bilayer systems. We also show that a model of tempered fractional Brownian motion recently proposed by Sabzikar and Meerschaert leads to physically very different behaviour with a seemingly paradoxical ballistic long time scaling.
Journal Article
A new DTAR (diversity–time–area relationship) model demonstrated with the indoor microbiome
Aim The spatio‐temporal distribution of biodiversity is a core field of biogeography, and the so‐termed species–time–area relationship (STAR), together with its siblings, that is the SAR (species–area relationship) and STR (species–time relationship), has achieved the rare status of classic laws in ecology and biogeography. Traditionally, the STAR or its recent generalization DTAR (diversity–time–area relationship) has been described with the bivariate power law (BPL) model or more recently with Whittaker, Triantis, and Ladle (2008, Journal of Biography; 35: 18) general dynamic model (GDM). We propose to extend the classic BPL into a more flexible DTAR model, which offers new quantitative methods for estimating maximal global diversity and charactering the relationship between local and regional diversity. Location Indoor microbiome. Taxon Microbes. Method We revise the BPL model by introducing two taper‐off (cut‐off) parameters or BPLEC (bivariate power law with exponential cutoffs) model, which eventually overwhelms the unsaturated increase of diversity over time and/or space and consequently can offer more realistic modelling of the joint spatio‐temporal distribution of biodiversity. Based on the BPLEC model, we further define three new concepts for DTAR: maximal accrual diversity (MAD) profile, local‐to‐regional diversity (LRD) ratio profile and local‐to‐global diversity (LGD) ratio profile. Results We introduce and demonstrate the new BPLEC model with the indoor microbiome datasets (Lax et al., 2014, Science; 345: 1048–1052). The new model fitted to the microbiome datasets equally well or slightly better than existing BPL and GDM models, but it possesses two advantages stated below. Main conclusion First, the new BPLEC model overcomes the unlimited diversity accrual in temporal and/or spatial dimensions and hence offers more realistic modelling to the DTAR. Second, the MAD and LRD/LGD offer useful methods for estimating the “dark” or “potential” diversity, which accounts for the species locally absent but present in a habitat‐specific regional species pool.
Journal Article
Power-Law Distributions in Empirical Data
Power-law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man-made phenomena. Unfortunately, the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution—the part of the distribution representing large but rare events—and by the difficulty of identifying the range over which power-law behavior holds. Commonly used methods for analyzing power-law data, such as least-squares fitting, can produce substantially inaccurate estimates of parameters for power-law distributions, and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all. Here we present a principled statistical framework for discerning and quantifying power-law behavior in empirical data. Our approach combines maximum-likelihood fitting methods with goodness-of-fit tests based on the Kolmogorov—Smirnov (KS) statistic and likelihood ratios. We evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches. We also apply the proposed methods to twenty-four real-world data sets from a range of different disciplines, each of which has been conjectured to follow a power-law distribution. In some cases we find these conjectures to be consistent with the data, while in others the power law is ruled out.
Journal Article