Suppose we select $n$ random points within a circle (independent, with continuous uniform distribution). Then we connect each point to its closest neighbor with an edge (note that "being the closest neighbor" is not necessarily a symmetric relation, i.e. the closest neighbor of the closest neighbor of a point $A$ is not necessarily the point $A$). The result is a graph that may have several connected components. Let's call the number of points in a connected component its size.
- What is the expected number of connected components in that graph?
What is the asymptotic behavior of that number when $n\to\infty\,?$ - What is the expected median size of those connected components?
What is the asymptotic behavior of that size when $n\to\infty\,?$ - What are the answers if instead of a circle (2-dimensional) we select random points within a sphere (3-dimensional)?
