Foundations of Data Science

edges between the two frontier sets is encountered is (1 − p) Ω(n4/3) ≤ e −Ω(dn1/3) , which
converges to zero. So almost surely, one of the edges is encountered and u and v end up
in the same connected component. This argument shows for a particular pair of vertices
u and v, the probability that they belong to different large connected components is very
small. Now use the union bound to conclude that this does not happen for any of the ( )
n
2
pairs of vertices. The details are left to the reader.
For the d < 1 case, almost surely, there is no connected component of size Ω(ln n).
Theorem 4.9 Let p=d/n with d < 1. The probability that G(n, p) has a component of
size more than c ln n is at most 1/n for a suitable constant c depending on d but not on
(1−d) 2
n.
Proof: There is a connected component of size at least k containing a particular vertex
v only if the breadth first search started at v has a nonempty frontier at all times up
to k. Let z k be the number of discovered vertices after k steps. The probability that
v is in a connected component of size greater than or equal to k is less than or equal
to Prob(z k > k). Now the distribution of z k is Binomial ( n − 1, 1 − (1 − d/n) k) . Since
(1 − d/n) k ≥ 1 − dk/n, the mean of Binomial ( n − 1, 1 − (1 − d/n) k) is less than the mean
of Binomial(n, dk n ). Since Binomial(n, dk n ) has mean dk, the mean of z k is at most dk where
d < 1. By a Chernoff bound, the probability that z k is greater than k is at most e −c 0k for
some constant c 0 > 0. If k ≥ c ln n for a suitably large c, then this probability is at most
1/n 2 . This bound is for a single vertex v. Multiplying by n for the union bound completes
the proof.
4.4 Branching Processes
As discussed in the previous section, a branching process is a method for creating a
random tree. Starting with the root node, each node has a probability distribution for
the number of its children. The root of the tree denotes a parent and its descendants are
the children with their descendants being the grandchildren. The children of the root are
the first generation, their children the second generation, and so on. Branching processes
have obvious applications in population studies, and as we saw earlier in exploring a connected
component in a random graph. Here, we give a more in-depth discussion of their
properties.
We analyze a simple case of a branching process where the distribution of the number
of children at each node in the tree is the same. The basic question asked is what is the
probability that the tree is finite, i.e., the probability that the branching process dies out?
This is called the extinction probability.
Our analysis of the branching process will give the probability of extinction, as well
as the expected size of the components conditioned on extinction. This will imply that in
G(n, d ), with d > 1, there is one giant component of size Ω(n), the rest of the components
n
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