# Questions tagged [power-law]

A power-law is a function that increases proportionally to a power of its argument (ax^b). Often seen in fitted relationships or in densities (power-law distributions).

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42 views

### Fit a function whose asymptotics is known

Consider the experimental blue curve in the figure that follows It turns out that I know the expression for the tails (-they are power laws- red and green curves) and I want to fit the curve using ...
24 views

### What is the range of $a$ such that $\frac{logN}{N^{3a+1/2}}\rightarrow 0$ as $N\rightarrow \infty$? [migrated]

I want to find the range of $a$ such that $\frac{logN}{N^{3a+1/2}}\rightarrow 0$ as $N\rightarrow\infty$. The answer to this question hinges on conditions under which $logN$ diverges slower than the ...
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### Fitting a lognormal-pareto distribution to empirical data distribution in R

Let's say I have data whose distribution is leptokurtic, with a lognormal peak and a pareto tail (for this question, I will generate lognormal-pareto data using package CompLognormal but I will need ...
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### Help with Power Curve MCMC

I'm trying to analyze some $D$ non-logistic cumulative data in a time series, bounded below by 0 and unbounded above. Splitting data into $W$ time windows of $d$ days, I know each window can be ...
18 views

### Power-Law Practical Consequences

What are the practical consequences/caveats when an analysed dataset is found to follow the power-law distribution? For example, I desire to perform cluster analysis on some control data whose ground ...
126 views

### Sampling distribution of the mean of the discrete-power law distribution

For a certain problem I wish to generate random integers $k$ so that their distribution follows $p_k \sim k^{-\alpha}$ for $k \geq k_{\text{min}}$, $k_{\text{min}} > 0$. I am following the ...
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### In the poweRlaw package, is the location parameter estimated for a lognormal distribution the $median=\exp(\mu)=\theta$ or $\mu$?

I have a set of graphs (each with the same nodes but with edges' weights defined by different research subjects) for which I would like to report statistics. One of these statistics is the betweenness ...
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### Why do poweRlaw and fitdistrplus differ in fitted lognorm parameters

I am trying to evaluate whether a power-law fit is appropriate for some data of lake areas that we have, and whether the theoretically supported alternative of the log-normal, at least at the tails, ...
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### Appropriate Data Analysis when Criterion has Heavy-Tailed Distribution

I have a data set where my independent variables (i.e., personality assessment scores) are continuous and follow a normal distribution. The criterion, sales performance, is heavy-tailed and follows a ...
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### Is there a named distribution with the property $P(X>10^k) = p^k$?

If I'm doing my math correctly, the exponential distribution has the property $P(X >k) = p ^ k$ (with $p$ conventionally written as $e ^ {-\lambda}$). I'm wondering if there is a different ...
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### How to interpret kernel coefficients in Hawkes Models

I am learning Hawkes Models application in Finance. I was reading the following article, which uses Tick for Python to estimate kernels as a power law. I could not understand what those values in ...
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### Why Are Bell Curves Used for Performance Appraisals?

Why do organizations that stack-rank employees force-fit employee rankings to a bell curve? Is there any theoretical or emperical evidence to support a normal or near-normal distribution of ...
268 views

### Cut off long tail of distribution (no matter how long is the tail) [closed]

One of the algorithms that we are working on is designed to report the time it takes for people to finish the onboarding process in an app. We need an algorithm to eliminate the long tail of a ...
23 views

### Validity of Monte-Carlo method to estimate a probability distribution which follows a power law

I am using a Monte-Carlo method to estimate a probability distribution function (pdf). Basically, I have several input parameters following known distributions, from which I can draw samples, that I ...
142 views

### How to perform the Kolmogorov-Smirnov test on the truncated power-law distributed data?

Recently, I read papers that perform power-law fitting on their empirical data (estimate the alpha), some of them report corresponding p-value for the Kolmogorov-Smirnov test, but many of them do not. ...
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### Powerlaw Distribution and old nodes [closed]

I am working on the size of discussions (number of replies, replies to the replies) on Twitter data, and I am observing Power-law distribution for small and medium-size discussion but it is not valid ...
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### Rank-size log-log plot inflection: drooping tail of power law

Most of my data seems nicely to fit a power law but with a "drooping tail", which I believe is quite common, although in this case the drop-off is quite steep. I have two related questions if I may ...
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### How to Select the Largest Y Values for {X,Y} Pairs, for a Pareto-Distributed Dataset, for a Meaningful Fit?

First, apologies for the inelegant question. Second, on to the question: Background Information: I study impact craters, and the size-frequency distribution (number vs diameter) of impact craters ...
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### Random Samples from Spliced Distribution

I am studying Clauset, Shalizi, and Newman, Power Law Distributions in Empirical Data (preprint available here) in R. Packages used: ...
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### How can I calculate the exponent (parameter or tail index) of the Pareto distribution associated to the following paperclips experiment?

The experiment is the following: take a pool of paperclips for which in each round we follow these steps: 1) choose two paperclips randomly 2) if they are not linked with each other then link them ...
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### Finding the exponents of a multiple power law: is linear regression valid?

I want to fit a multiple power law equation of the form $y = {x_1}^{\alpha_1} {x_2}^{\alpha_2}$ where I have many examples of $y, x_1, x_2$. (Note there is no intercept.) Is it possible for me to ...
304 views

### Discrete Pareto Distribution vs Zipf Distribution and Power Law vs Zipf Law

I need to get a simple, but clear idea of Discrete Pareto Distribution vs Zipf Distribution and Power Law vs Zipf Law. (Are they similar/ how they relate to each other.) Wikipedia definitions do not ...
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### How are these two power law fitting glm models different?

I have some data that I thought I'd try fitting with a power law (in R). ...
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### Using log-log graph to find equation of power law relationship?

I have a set of data that I think forms a power law relationship, however I am struggling to work out the equation of the relationship. Here is a subset of the data I am working with: ...
109 views

### Sum of powers of standard normal random variables

Context: While trying to teach the Central Limit Theorem I thought it would be a good idea to show a case where it breaks down. Question: Consider the sum of increasing powers of standard normal ...
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### How well is a power law distribution described by the first four moments?

For a normal distribution, the first two moments (mean and variance) are sufficient statistics for the entire distribution. Suppose I have a power law distribution, and I have data on the first, ...
281 views

### Is the truncated power law a heavy-tailed distribution?

A heavy-tailed distribution is often defined as a distribution with a tail that is not exponentially bounded. A truncated power law (or power law with exponential cut-off) is a distribution that ...
376 views

### Most accurate way to express the income of the top 1% in a power distribution?

I have an array of 50,972 household incomes for a small metro area, and I want to measure how much the so-called top "one-percent" make (e.g., the 99th percentile). As you'd expect, it's a power curve:...
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### Is KS test really appropriate when validating a power law/estimating power law parameters?

I'm attempting to find out whether some highly skewed data are drawn from a power law distribution, following the popular paper by Clauset, Shalizi and Newman, 2009. Clauset et al. use the Kolmogorov-...
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### Connection between logit model and power law distribution

I am wondering whether it is appropriate to make a connection between the use of a logit model and a power law distribution. My dependent variable is categorical and ordinal, therefore I am fitting a ...
554 views

### Calculate Zipf-Mandelbrot parameters from distribution

I am fetching trending topics from social media where the frequency of likes is said to follow a Zipf-Mandelbrot distribution; i.e., some of the posts will have a high number of likes and some other ...
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### Determining the number of observations within the tail of a given distribution

I am wondering how to determine the number of observations that fall within the tail of a distribution. I am reading a paper and the authors use the assumption that 50 observations need to fall into ...
1k views

### Characterizing/Fitting Word Count Data into Zipf / Power Law / LogNormal

Using NLTK and Pandas, I was able to process some text files and generate word count data for them, and finally create a histogram describing word frequency. However, I'm wondering what kind of ...
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### Extracting power of a power law from data

My question is more about the methodology. Assuming in some experiment we have measured quantity $y$ per each unit of time $x.$ So $y$ and $x$ form our data set here. Moreover, we know that they are ...
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### “Law of large numbers” for distribution with infinite variance?

This is a purely explorative question. I asked a question here about a "central limit theorem" for random variables with infinite variance. I did not expect it, but it turns out that even some ...
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### A random variable $X$ on $(0,\infty)$ which behaves like Exp for small $x$ and Pareto for large $x$

Are there any examples of distributions which behave like Exponential for small values and like Pareto for large values. \ln \mathbb{P}[X>x] \sim -\lambda x, \qquad \text{ for } x \text{ small}, ...
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### Mixture of power-law distribution

I have two large sets A and B of (integer) numbers, both obtained with two different (and unknown) probability distributions $\rho_A$ and $\rho_B$. A (also large) set C contains a proportion $p$ of ...
1k views

### Normalization of power-law distributed variables. Z-scores or Min-Max?

I need to make a composite index from the sum of three power-law distributed variables, which vary on different scales and have different variances. For each variable there are many observations with ...
487 views

### Generate random numbers from a power-law/exponential distribution

I'm working through the paper Power-law distributions in empirical data by Clauset, Newman and Shalizi. On page 12, they generate random numbers from the following distribution: \begin{equation*} p(...
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### Generating random samples from a power law and testing them with R igraph

I haven't looked into power laws before, so the question may very well have an obvious, embarrassing answer. In some ways it dovetails with this post. The problem may be in the method, or in the ...
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### What is a hooked powerlaw?

I happened to encounter lots of scientific/business scenarios where a Zipf/Pareto/powerlaw describes well my data. However, whenever the mean of the distribution is large enough, the fact that these ...
605 views

### Fitting a Pareto distribution to two dimensional data in R

I've looked here: How do I fit a set of data to a Pareto distribution in R? do-i-fit-a-set-of-data-to-a-pareto-distribution-in-r and I've checked out the poweRlaw library which is built for fitting ...
895 views

### Exponential vs Power Regression - which one is more appropriate when data points are limited?

When researching the reciprocity failure (a feature of film photography, otherwise more appropriate to https://photo.stackexchange.com/) I have run into a statistical issue. I have a datasheet for a ...
3k views

### Estimate power law exponent for node degree distribution in scale free networks

I am trying to use the powerlaw python package to estimate the power law exponent of the degree distribution in a graph. As a reference I am using ...
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### Input to fit a power-law to degree distribution of a network

I would like to use R to test whether the degree distribution of a network behave like a power-law with scale-free property. Nonetheless, I've read diferent people ...
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### Log-likelihood in fit_power_law{igraph}

The R package igraph has the fit_power_law function which, as you can imagine, can fit a ...
2k views

### Fitting a power law to the relationship between two variables

I have a small set of data that I need to fit a curve to (see image and data below). ...