R: Fitting a power-law distribution function to discrete data

fit_power_law {igraph} | R Documentation

## Fitting a power-law distribution function to discrete data

### Description

`fit_power_law` fits a power-law distribution to a data set.

### Usage

    fit_power_law(x, xmin = NULL, start = 2, force.continuous = FALSE,
      implementation = c("plfit", "R.mle"), ...)

### Arguments

`x` | The data to fit, a numeric vector. For implementation ‘`R.mle`’ the data must be integer values. For the ‘`plfit`’ implementation non-integer values might be present and then a continuous power-law distribution is fitted.
`xmin` | Numeric scalar, or `NULL`. The lower bound for fitting the power-law. If `NULL`, the smallest value in `x` will be used for the ‘`R.mle`’ implementation, and its value will be automatically determined for the ‘`plfit`’ implementation. This argument makes it possible to fit only the tail of the distribution.
`start` | Numeric scalar. The initial value of the exponent for the minimizing function, for the ‘`R.mle`’ implementation. Ususally it is safe to leave this untouched.
`force.continuous` | Logical scalar. Whether to force a continuous distribution for the ‘`plfit`’ implementation, even if the sample vector contains integer values only (by chance). If this argument is false, igraph will assume a continuous distribution if at least one sample is non-integer and assume a discrete distribution otherwise.
`implementation` | Character scalar. Which implementation to use. See details below.
`...` | Additional arguments, passed to the maximum likelihood optimizing function, `mle`, if the ‘`R.mle`’ implementation is chosen. It is ignored by the ‘`plfit`’ implementation.

### Details

This function fits a power-law distribution to a vector containing samples from a distribution (that is assumed to follow a power-law of course). In a power-law distribution, it is generally assumed that P(X=x) is proportional to x^-alpha, where x is a positive number and alpha is greater than 1. In many real-world cases, the power-law behaviour kicks in only above a threshold value xmin. The goal of this function is to determine alpha if xmin is given, or to determine xmin and the corresponding value of alpha.

`fit_power_law` provides two maximum likelihood implementations. If the `implementation` argument is ‘`R.mle`’, then the BFGS optimization (see mle) algorithm is applied. The additional arguments are passed to the mle function, so it is possible to change the optimization method and/or its parameters. This implementation can not to fit the xmin argument, so use the ‘`plfit`’ implementation if you want to do that.

The ‘`plfit`’ implementation also uses the maximum likelihood principle to determine alpha for a given xmin; When xmin is not given in advance, the algorithm will attempt to find itsoptimal value for which the p-value of a Kolmogorov-Smirnov test between the fitted distribution and the original sample is the largest. The function uses the method of Clauset, Shalizi and Newman to calculate the parameters of the fitted distribution. See references below for the details.

### Value

Depends on the `implementation` argument. If it is ‘`R.mle`’, then an object with class ‘`mle`’. It can be used to calculate confidence intervals and log-likelihood. See `mle-class` for details.

If `implementation` is ‘`plfit`’, then the result is a named list with entries:

`continuous` | Logical scalar, whether the fitted power-law distribution was continuous or discrete.
`alpha` | Numeric scalar, the exponent of the fitted power-law distribution.
`xmin` | Numeric scalar, the minimum value from which the power-law distribution was fitted. In other words, only the values larger than `xmin` were used from the input vector.
`logLik` | Numeric scalar, the log-likelihood of the fitted parameters.
`KS.stat` | Numeric scalar, the test statistic of a Kolmogorov-Smirnov test that compares the fitted distribution with the input vector. Smaller scores denote better fit.
`KS.p` | Numeric scalar, the p-value of the Kolmogorov-Smirnov test. Small p-values (less than 0.05) indicate that the test rejected the hypothesis that the original data could have been drawn from the fitted power-law distribution.

### Author(s)

Tamas Nepusz ntamas@gmail.com and Gabor Csardi csardi.gabor@gmail.com

### References

Power laws, Pareto distributions and Zipf's law, M. E. J. Newman, Contemporary Physics, 46, 323-351, 2005.

Aaron Clauset, Cosma R .Shalizi and Mark E.J. Newman: Power-law distributions in empirical data. SIAM Review 51(4):661-703, 2009.

### See Also

`mle`

### Examples

    # This should approximately yield the correct exponent 3
    g <- barabasi.game(1000) # increase this number to have a better estimate
    d <- degree(g, mode="in")
    fit1 <- fit_power_law(d+1, 10)
    fit2 <- fit_power_law(d+1, 10, implementation="R.mle")

    fit1$alpha
    stats4::coef(fit2)
    fit1$logLik
    stats4::logLik(fit2)

[Package igraph version 1.2.2 Index]