Foundations of Data Science

For a small number i of steps, the probability distribution of the size of the set of
discovered vertices at time i is p(k) = e −di (di) k
and has expected value di. Thus, the
k!
expected size of the frontier is (d − 1)i. For the frontier to be empty would require that
the size of the set of discovered vertices be smaller than its expected value by (d − 1)i.
That is, the size of the set of discovered vertices would need to be di − (d − 1)i = i. The
probability of this is
e −di (di) i
= e −di di i i
e i = e −(d−1)i d i −(d−1−ln d)i
= e
i! i i
which drops off exponentially fast with i provided d > 1. Since d − 1 − ln d is some
constant c > 0, the probability is e −ci which for i = ln n is e −c ln n = 1 n c . Thus, with high
probability, the largest small component in the graph is of size at most ln n.
Illustration 4.1
at most t − 1 is exactly a i t−1. With this we get:
a t = p 0 +
s∑
p i a i t−1 =
i=1
s∑
p i a i t−1.
i=0
We have a 1 = p 0 < 1. There is a constant α ∈ [p 0 , 1) such that whenever a t−1 ≤ α, the
above recursion implies that a t ≤ α. This would finish the proof since then a 1 ≤ α implies
a 2 ≤ α which implies a 3 ≤ α etc. and so q = lim t→∞ a t ≤ α.
To prove the claim, consider the polynomial
h(x) = x −
s∑
p i x i .
i=0
We see that h(1) = 0 and h ′ (1) = 1 − ∑ s
i=1 ip i ≈ 1 − sd , which is at most a strictly
n
negative constant. By continuity of h(·), there exists some x 0 < 1 such that h(x) ≥ 0 for
x ∈ [x 0 , 1]. Take α = Max(x 0 , p 0 ). Now since ∑ s
i=0 p ix i has all nonnegative coefficients,
it is an increasing function of x and so if a t−1 is at least α, then, ∑ s
∑ i=0 p ia i t−1 is at least
s
i=0 p iα i ≥ α. Now, if a t−1 ≤ α,
proving the claim.
a t =
s∑
p i a i t−1 ≥
i=0
s∑
p i α i = α − h(α) ≤ α,
i=1
We now prove in Theorem 4.8 that in G(n, d ), d > 1, there is one giant component
n
containing a fraction of the n vertices and that the remaining vertices are in components
of size less than some constant c 1 times log n. There are no components greater than
c 1 log n other than the giant component.
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