$n$ points ($P_i$) are distributed uniformly on the surface of an unit radius sphere. 2 points are interconnected if the distance between them is $\le r$ (thresold). We call the degree of point $i$ ($d_i$) the number of edge (interconnexion between 2 points) coming from that point. The expected average of degrees for all the points is $d$. What is the formula for $r$ to get an average degree close to $d$?
I found a solution for this problem for an unit square here : Average Degree of a Random Geometric Graph
It is described there that the expected average is $(N−1)p(\delta)$ where $p(\delta)=P(\|X-Y\|\leqslant\delta)$ for some i.i.d. $X$ and $Y$ uniformly distributed in the square.
The expression of $p(δ)$ is discussed here http://mathworld.wolfram.com/SquareLinePicking.html
I think that my problem is similar but here the problem is to find the average distance between two points picked in the surface of a sphere.
I found one demonstration here : http://arxiv.org/abs/math-ph/0004021 (page 5 & 6) Is it the expression of the probability $p(\delta)$ for a sphere with uniform distribution of points?
Starting with http://mathworld.wolfram.com/SphereLinePicking.html.
Set $P_1=(0,0,1)$
Then $P_2=(\sqrt{1-u^2}\cos\theta,\sqrt{1-u^2}\sin\theta,u)$
From http://mathworld.wolfram.com/SphericalDistance.html
$l=\cos^{-1}(P_1.P_2)=\cos^{-1}u$
This is an http://en.wikipedia.org/wiki/Arcsine_distribution with $a=-1$ and $b=1$ and has a probability density function of:
$$f(x)=\frac{1}{\pi\sqrt{(1+x)(1-x)}}$$
And a cummulative distribution function of:
$$F(x)=\frac{2}{\pi}\arcsin\frac{x+1}{2}$$
Can you continue from there?