igraph (version 1.0.0)

fit_power_law: 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 distr

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 determ

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 distribut

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

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.

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 $x_{min}$. The goal of this function is to determine $\alpha$ if $x_{min}$ is given, or to determine $x_{min}$ 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 $x_{min}$ 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 $x_{min}$; When $x_{min}$ 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.

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)