I'm trying to figure out a power law distribution using Facebook friends as an example. Using data from 2011, supposing all users have friends in a range from 20 to 5000: the probability for each person to have d friends is p(d)=C/(d^2.2).
So, I estimated C numerically using the following equation (because the constant C is determined by the condition that the sum of probabilities from d=20 to d=5000 is equal to 1):
1 = C/(20^2.2) + C/(5000^2.2).
The probability that a vertex has degree 5000 is then
p(d) = 728.226/5000^2.2 = .000005303
I want to now find out how I would estimate the expected number of vertices with the degree 5000. Your help is much appreciated, thanks!
"does my calculation of C make sense? the probability of the min + max of the range equal to one?" Not quite, for your model, you need to set $C$ so that $p$ is a probability distribution, that is, you set $C$ so that $$\sum_{d=20} ^{5000} C/d^{2.2} =1$$ Which is the same as setting $C$ so that $$C=\frac{1}{\sum_{d=20} ^{5000} 1/d^{2.2} }$$ I plugged this sum into Mathematica and got: $$\sum_{d=20}^{5000} \frac{1}{d^{2.2}} =0.0235555$$ So, you have $$C=42.4529$$
To answer your other question *"... so 5000 * .000005303":*
this is not correct, unless your sample set had exactly 5000 users. Given a group of $n$ users, the expected number of users with 5000 friends would be $$n \cdot p(5000) = n\cdot (3.09153 \cdot 10^{-7})$$
You were pretty close though! Hope this helps.