Foundations of Data Science

Theorem 4.8 Let p=d/n with d > 1.
1. There are constants c 1 and c 2 such that the probability that there is a connected
component of size between c 1 log n and c 2 n is at most 1/n.
2. The number of vertices in components of size O(ln n) is almost surely at most cn
for some c < 1. Thus, with probability 1 − o(1), there is a connected component of
size Ω(n).
3. The probability that there are two or more connected components, each of size more
than n 2/3 , is at most 1/n.
Proof: We begin with (1). If a particular vertex v is in a connected component of size k,
then the frontier of the breadth first search process started at v becomes empty for the first
time at time k implying that z k = k. Now z k has distribution Binomial ( n − 1, 1 − (1 − (d/n)) k) .
Let c 2 = d−1 and assume k < c
d 2
2 n and k > c 1 log n for a sufficiently large c 1 . By truncated
Taylor series expansion, it is easy to see that
(1 − d n )k ≤ 1 − kd
n + k2 d 2
2n 2 .
Thus, E(z k ) ≥ kd − (k 2 d 2 /2n). So, if z k were to be exactly k, then its deviation from its
mean is at least kd − (k 2 d 2 /(2n)) − k ≥ (k(d − 1)/2) ≥ Ω(k). So by a Chernoff bound
Prob(z k = k) ≤ e −Ω(k) . If k is greater than c 1 log n for some sufficiently large c 1 , then
z k = k with probability at most 1/n 3 and this along with a union bound gives (1).
Next consider (2). For a vertex v, let cc(v) denote the set of vertices in the connected
component containing v. By (1), almost surely, cc(v) is a small set of size at most c 1 log n
or a large set of size at least c 2 n for every vertex v. The central part of the proof of
(2) that the probability of a vertex being in a small component is strictly less than one
was established in Lemma 4.7. Let x be the number of vertices in a small connected
component. Lemma 4.7 implies that the expectation of the random variable x equals the
number of vertices in small connected components is at most some c 3 n, for a constant c 3
strictly less than one. But we need to show that for any graph almost surely the actual
number x of such vertices is at most some constant strictly less than one times n. For
this, we use the second moment method. In this case, the proof that the variance of x is
o(E 2 (x)) is easy. Let x i be the indicator random variable of the event that cc(i) is small.
Let S and T run over all small sets. Noting that for i ≠ j, cc(i) and cc(j) either are the
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