From 3aee2fd43e3059a699af2b63c6f2395e5a55e515 Mon Sep 17 00:00:00 2001
From: KatolaZ <katolaz@freaknet.org>
Date: Wed, 27 Sep 2017 15:06:31 +0100
Subject: First commit on github -- NetBunch 1.0

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+fitmle(1) -- Fit a set of values with a power-law distribution
+======
+
+## SYNOPSIS
+
+`fitmle` <data_in> [<tol> [TEST [<num\_test>]]]
+
+## DESCRIPTION
+
+`fitmle` fits the data points contained in the file <data_in> with a
+power-law function P(k) ~ k^(-gamma), using the Maximum-Likelihood
+Estimator (MLE). In particular, `fitmle` finds the exponent `gamma`
+and the minimum of the values provided on input for which the
+power-law behaviour holds. The second (optional) argument <tol> sets
+the acceptable statistical error on the estimate of the exponent.
+
+If `TEST` is provided, the program associates a p-value to the
+goodness of the fit, based on the Kolmogorov-Smirnov statistics
+computed on <num_test> sampled distributions from the theoretical
+power-law function. If <num_test> is not provided, the test is based
+on 100 sampled distributions.
+
+
+## PARAMETERS
+
+* <data_in>:
+    Set of values to fit. If is equal to `-` (dash), read the set from
+    STDIN.
+
+* <tol>: 
+    The acceptable statistical error on the estimation of the
+    exponent. If omitted, it is set to 0.1.
+    
+* TEST:
+    If the third parameter is `TEST`, the program computes an estimate
+    of the p-value associated to the best-fit, based on <num_test>
+    synthetic samples of the same size of the input set.
+
+* <num_test>:
+    Number of synthetic samples to use for the estimation of the
+    p-value of the best fit.
+
+## OUTPUT
+
+If `fitmle` is given less than three parameters (i.e., if `TEST` is
+not specified), the output is a line in the format:
+
+        gamma k_min ks
+
+where `gamma` is the estimate for the exponent, `k_min` is the
+smallest of the input values for which the power-law behaviour holds,
+and `ks` is the value of the Kolmogorov-Smirnov statistics of the
+best-fit. 
+
+If `TEST` is specified, the output line contains also the estimate of
+the p-value of the best fit:
+
+        gamma k_min ks p-value
+
+where `p-value` is based on <num_test> samples (or just 100, if
+<num_test> is not specified) of the same size of the input, obtained
+from the theoretical power-law function computed as a best fit.
+ 
+## EXAMPLES
+
+Let us assume that the file `AS-20010316.net_degs` contains the degree
+sequence of the data set `AS-20010316.net` (the graph of the Internet
+at the AS level in March 2001). The exponent of the best-fit power-law
+distribution can be obtained by using:
+
+        $ fitmle AS-20010316.net_degs 
+        Using discrete fit
+        2.06165 6 0.031626 0.17
+        $
+
+where `2.06165` is the estimated value of the exponent `gamma`, `6` is
+the minimum degree value for which the power-law behaviour holds, and
+`0.031626` is the value of the Kolmogorov-Smirnov statistics of the
+best-fit. The program is also telling us that it decided to use the
+discrete fitting procedure, since all the values in
+`AS-20010316.net_degs` are integers. The latter information is printed
+to STDERR.
+
+It is possible to compute the p-value of the estimate by running:
+
+        $ fitmle AS-20010316.net_degs 0.1 TEST
+        Using discrete fit
+        2.06165 6 0.031626 0.17
+        $
+
+which provides a p-value equal to 0.17, meaning that 17% of the
+synthetic samples showed a value of the KS statistics larger than that
+of the best-fit. The estimation of the p-value here is based on 100
+synthetic samples, since <num_test> was not provided. If we allow a
+slightly larger value of the statistical error on the estimate of the
+exponent `gamma`, we obtain different values of `gamma` and `k_min`,
+and a much higher p-value:
+
+        $ fitmle AS-20010316.net_degs 0.15 TEST 1000
+        Using discrete fit
+        2.0585 19 0.0253754 0.924
+        $
+
+Notice that in this case, the p-value of the estimate is equal to
+0.924, and is based on 1000 synthetic samples.
+
+## SEE ALSO
+
+deg_seq(1), power_law(1)
+
+
+## REFERENCES
+
+* A\. Clauset, C. R. Shalizi, and M. E. J. Newman. "Power-law
+  distributions in empirical data". SIAM Rev. 51, (2007), 661-703.
+
+* V\. Latora, V. Nicosia, G. Russo, "Complex Networks: Principles,
+  Methods and Applications", Chapter 5, Cambridge University Press
+  (2017)
+
+
+
+## AUTHORS
+
+(c) Vincenzo 'KatolaZ' Nicosia 2009-2017 `<v.nicosia@qmul.ac.uk>`.
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