uj_ wrote:I was surprised as well not to find a discrete version of this distribution, at least somewhere. Kleinberg's results are interesting, that in small world networks the optimal random link distribution is 1/d^r, where d is the distance between the nodes, and r is effectively the number of dimensions in the network. The r=1 case is interesting in how it breaks calculations for existing power law distributions, probably because integrating 1/x is a special case from the general 1/x^r(?) The Harmonic Series is also interesting in that if you remove terms that have specific digits in it, like 9, suddenly it doesn't diverge anymore.phineasgage wrote:I haven't been doing calculus for a while but I suddently realized that this,
The continuous distribution from uj_ works great. Although it isn't perfect in the discrete case, it can still be used if the size of N is large and the application doesn't demand perfect accuracy in the lower range.
X = e^[ln(N)*U]
can be further reduced to,
X = [e^[ln(N)]]^U = N^U
This means that the pow method can be used for the simulation code, like
Considering that this distribution is so easy to simulate (in the continous case) it's quite astonishing that there are so few references to it. I haven't found a single one.
Random rand = new Random(); int n = 100; double dU = rand.nextDouble(); double dX = Math.pow(n, dU); // dX is the random number on the distribution
And the distribution is interesting. Every number in the range 1 to N comes up with a probability that's proportional to the inverse of the number. Informally this corresponds to the notion that "the shorter/smaller/sooner the more probable, the longer/larger/later the less probable". This should be a common situation in "nature". Doesn't stuff tend to split up in many small and few large? Doesn't people prefer sooner rather than later?
Further the distribution has a "fat tail". This means that numbers far away from the average still has a substantial probability of occuring. This is not the case in the Gaussian/Normal distribution for example where probabilities quickly get extremely small (like not even once in the lifetime of the universe).
The distribution also is mathematically truncated. It's limited to a specific range. Again, this is not the case with the Gaussian/Normal distribution. It must be artificially truncated in principle skewing it.
Now what should this good cigarr be called? Well I suggest the Truncated Harmonic Distribution until someone manages to Google up its given name -:)