Foundations of Data Science

The probability that the long distance edge from u goes to v is proportional to
d −r (u, v). Note that the constant of proportionality will vary with the vertex u depending
on where u is relative to the border of the n × n grid. However, the number of
vertices at distance exactly k from u is at most 4k and for k ≤ n/2 is at least k. Let
c r (u) = ∑ v d−r (u, v) be the normalizing constant. It is the inverse of the constant of
proportionality.
For r > 2, c r (u) is lower bounded by
c r (u) = ∑ v
n/2
n/2
∑ ∑
d −r (u, v) ≥ (k)k −r = k 1−r ≥ 1.
k=1
k=1
No matter how large r is the first term of ∑ n/2
k=1 k1−r is at least one.
For r = 2 the normalizing constant c r (u) is upper bounded by
c r (u) = ∑ v
d −r (u, v) ≤
2n∑
k=1
(4k)k −2 ≤ 4
2n∑
k=1
1
k
= θ(ln n).
For r < 2, the normalizing constant c r (u) is lower bounded by
c r (u) = ∑ v
n/2
∑
d −r (u, v) ≥ (k)k −r ≥
k=1
n/2
∑
k=n/4
k 1−r .
The summation
n/2
∑
k=n/4
k 1−r has n 4 terms, the smallest of which is ( n
4
) 1−r
or
( n
2
) 1−r
depending
on whether r is greater or less than one. This gives the following lower bound on c r (u).
c r (u) ≥ n 4 ω(n1−r ) = ω(n 2−r ).
No short paths exist for the r > 2 case.
For r > 2, we first show that for at least half of the pairs of vertices, there is no short
path between them. We begin by showing that the expected number of edges of length
greater than n r+2
2r goes to zero. The probability of an edge from u to v is d −r (u, v)/c r (u)
where c r (u) is lower bounded by a constant. Thus the probability that a particular edge
of length greater than or equal to n r+2
2r
( )
is chosen is upper bounded by some contant c
−r
times n r+2
2r or cn
−( r+2
2 ) . Since there are n 2 long edges, the expected number of edges
of length at least n r+2
2r is at most cn 2 n − (r+2)
2 or cn 2−r
2 , which for r > 2 goes to zero. Thus,
by the first moment method, almost surely, there are no such edges.
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