`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. Usually 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,

`stats4::mle()`

, if the ‘`R.mle`

’ implementation is chosen. It is ignored by the ‘`plfit`

’ implementation.

## 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
`stats4::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 \(x_\text{min}\). The goal of this function is to determine \(\alpha\) if \(x_\text{min}\) is given, or to determine \(x_\text{min}\) 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 \(x_\text{min}\) 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 \(x_\text{min}\);
When \(x_\text{min}\) 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.

## Author

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

## Examples

```
# This should approximately yield the correct exponent 3
g <- sample_pa(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$alpha
#> [1] 2.567177
stats4::coef(fit2)
#> alpha
#> 2.566887
fit1$logLik
#> [1] -58.53319
stats4::logLik(fit2)
#> 'log Lik.' -58.53288 (df=1)
```