Foundations of Data Science

Destination
Figure 4.17: For d < 2, ends points of long distance edges are uniformly distributed and
on a short path it is unlikely to encounter one that takes you close to the destination.
r > 2 The lengths of long distance edges tend to be short so the
probability of encountering a sufficiently long, long-distance edge is
too low.
r = 2 Selecting the edge with end point closest to the destination
finds a short path.
r < 2 The ends of long distance edges tend to be uniformly distributed.
Short paths exist but a polylog length path is unlikely
to encounter a long distance edge whose end point is close to the
destination.
Figure 4.18: Effects of different values of r on the expected length of long distance edges
and the ability to find short paths.
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