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Posted on May 30, 2008 @ 03:31:42 PM by Paul Meagher
Bernardo Huberman formulated a few "laws of the web". One of these laws concerns the distribution of clickstream lengths.
The probability density function of L, the length of a clickstream, is, according to the "Law of Surfing", distributed according to an inverse guassian function:
f(L, v, λ) = sqrt(λ/2π) * L-3/2 * e-(λ/2*v*v*L)*(L-v)^2
What do the symbols L, v, λ mean?
L is session length measured in page clicks.
v is the expected value or mean. A fitted distribution of path length probabilities for Boston University students had a value of v=51.19 which the
study referred to as "mean visits".
λ is related to the expected value and variance
λ = v3/Var(L). The fitted value for λ was equal to 3.53.
Here is a php script to compute the probability densities for each clickstream length:
<?php
/**
* Raises a number to a floating point power
*
* a^b = e^(b log a) which is not 10log but the e-log (aka "ln")
* so instead of pow( $a , 0.6 ) use something like
* exp( 0.6 * log($a))
*
* @see http://ca3.php.net/manual/en/function.pow.php#47297
*/
function fpow($base, $fexp) {
return exp($fexp * log($base));
}
/**
* Used to compute probability density associated with each clickstream
* length.
*/
function inverse_guassian($L, $v, $lambda) {
return sqrt($lambda/(2 * M_PI))
* fpow($L, -3/2)
* exp(-1.0 * ($lambda/(2 * pow($v, 2) * $L)) * pow($L-$v, 2));
}
$v = 51.19;
$lambda = 3.53;
for($L=1; $L < 20; $L++) {
$p = inverse_guassian($L, $v, $lambda);
echo "f(".$L.", ".$v.", ".$lambda.") = ".$p."<br />";
}
?>
And these are the probability densites the program generates:
f(1, 51.19, 3.53) = 0.13738001323331
f(2, 51.19, 3.53) = 0.11731429052801
f(3, 51.19, 3.53) = 0.08563996238308
f(4, 51.19, 3.53) = 0.064395173972196
f(5, 51.19, 3.53) = 0.050294705235689
f(6, 51.19, 3.53) = 0.040551681574111
f(7, 51.19, 3.53) = 0.033538749822867
f(8, 51.19, 3.53) = 0.028310951045661
f(9, 51.19, 3.53) = 0.024298496373469
f(10, 51.19, 3.53) = 0.021143050105347
f(11, 51.19, 3.53) = 0.018610350025871
f(12, 51.19, 3.53) = 0.016541931634992
f(13, 51.19, 3.53) = 0.014827372300491
f(14, 51.19, 3.53) = 0.013387711569616
f(15, 51.19, 3.53) = 0.012165196593551
f(16, 51.19, 3.53) = 0.0111167387119
f(17, 51.19, 3.53) = 0.0102096203282
f(18, 51.19, 3.53) = 0.0094186073802048
f(19, 51.19, 3.53) = 0.0087239635554072
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