power.law.fit {igraph}  R Documentation 
power.law.fit
fits a powerlaw distribution to a
data set.
power.law.fit(x, xmin=NULL, start=2, force.continuous=FALSE, implementation=c("plfit", "R.mle"), ...)
x 
The data to fit, a numeric vector. For implementation
‘ 
xmin 
Numeric scalar, or 
start 
Numeric scalar. The initial value of the exponent for the
minimizing function, for the ‘ 
force.continuous 
Logical scalar. Whether to force a continuous
distribution for the ‘ 
implementation 
Character scalar. Which implementation to use. See details below. 
... 
Additional arguments, passed to the maximum likelihood
optimizing function, 
This function fits a powerlaw distribution to a vector containing samples from a distribution (that is assumed to follow a powerlaw of course). In a powerlaw 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 realworld cases, the powerlaw 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.
power.law.fit
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
pvalue of a KolmogorovSmirnov 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.
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 loglikelihood. See
mleclass
for details.
If implementation
is ‘plfit
’, then the result is
a named list with entries:
continuous 
Logical scalar, whether the fitted powerlaw distribution was continuous or discrete. 
alpha 
Numeric scalar, the exponent of the fitted powerlaw distribution. 
xmin 
Numeric scalar, the minimum value from which the powerlaw
distribution was fitted. In other words, only the values larger than

logLik 
Numeric scalar, the loglikelihood of the fitted parameters. 
KS.stat 
Numeric scalar, the test statistic of a KolmogorovSmirnov test that compares the fitted distribution with the input vector. Smaller scores denote better fit. 
KS.p 
Numeric scalar, the pvalue of the KolmogorovSmirnov test. Small pvalues (less than 0.05) indicate that the test rejected the hypothesis that the original data could have been drawn from the fitted powerlaw distribution. 
Tamas Nepusz ntamas@gmail.com and Gabor Csardi csardi.gabor@gmail.com
Power laws, Pareto distributions and Zipf's law, M. E. J. Newman, Contemporary Physics, 46, 323351, 2005.
Aaron Clauset, Cosma R .Shalizi and Mark E.J. Newman: Powerlaw distributions in empirical data. SIAM Review 51(4):661703, 2009.
# 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 < power.law.fit(d+1, 10) fit2 < power.law.fit(d+1, 10, implementation="R.mle") fit1$alpha coef(fit2) fit1$logLik logLik(fit2)