Number of connected components of a random nearest neighbor graph? Let us sample some big number N points randomly uniformly on $[0,1]^d$. 
Consider  1-nearest neighbor graph based on such data cloud. (Let us look on it as UNdirected graph).

Question What would the number of connected components depending on d ? 
(As a percent of "N" - number of points.)

Simulation below suggest 31% for d=2,   20% for d=20, etc:

Percent Dimension:
31 2
28 5
25 10
20 20
15 50
13 100
10 1000


See code below. (One can run it on colab.research.google.com without installing anything on your comp). 

If someone can comment on more general questions here: 
https://mathoverflow.net/q/362721/10446
that would be greatly appreciated.

!pip install python-igraph
!pip install cairocffi
import igraph

import time
from sklearn.neighbors import NearestNeighbors
import numpy as np

t0 = time.time()

dim = 20
n_sample = 10**4

for i in range(10): # repeat simulation 10 times to get more stat
  X = np.random.rand(n_sample, dim)
  nbrs = NearestNeighbors(n_neighbors=2, algorithm='ball_tree', ).fit(X)
  distances, indices = nbrs.kneighbors(X)
  g = igraph.Graph( directed = True )
  g.add_vertices(range(n_sample))
  g.add_edges(indices )
  g2 = g.as_undirected(mode = 'collapse')
  r = g2.clusters()
  print(len(r),len(r)/n_sample*100 , time.time() - t0)
For $n$ uniformly random points in a unit square the number of components is

$$\frac{3\pi}{8\pi+3\sqrt{3}}n+o(n)$$

See Theorem 2 of D. Eppstein, M. S. Paterson, and F. F. Yao (1997), "On nearest-neighbor graphs", Disc. Comput. Geom. 17: 263–282, https://www.ics.uci.edu/~eppstein/pubs/EppPatYao-DCG-97.pdf

For points in any fixed higher dimension it is $\Theta(n)$; I don't know the exact constant of proportionality but the paper describes how to calculate it.