fit_power_law fits a power-law distribution to a data set.
fit_power_law(x, xmin = NULL, start = 2, force.continuous = FALSE,
implementation = c("plfit", "R.mle"), ...)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.
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.
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.
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.
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.
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:
Logical scalar, whether the fitted power-law distribution was continuous or discrete.
Numeric scalar, the exponent of the fitted power-law distribution.
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.
Numeric scalar, the log-likelihood of the fitted parameters.
Numeric scalar, the test statistic of a Kolmogorov-Smirnov test that compares the fitted distribution with the input vector. Smaller scores denote better fit.
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.
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.
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.
# NOT RUN {
# 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)
# }
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