Foundations of Data Science

Lemma 4.7 Assume d > 1. For any vertex v, the probability that cc(v), the connected
component containing v, is small (i.e., of size O(log n)) is a constant strictly less than 1.
Proof: Let p be the probability that cc(v) is small, i.e., the probability that the breadth
first search started at v terminates before c 1 log n vertices are discovered. Slightly modify
the breadth first search as follows: If in exploring a vertex u at some point, there are m
undiscovered vertices, choose the number k of vertices which will be adjacent to u from
the Binomial(m, d ) distribution. Having picked k, pick one of the ( )
m
n k subsets of k undiscovered
vertices to be the set of vertices adjacent to u, and make the other m − k vertices
not adjacent to u. This process has the same distribution as picking each edge from u
independently at random to be present with probability d/n. As the search proceeds, m
decreases. If cc(v) is small, m, the number of undiscovered vertices, is always greater than
s = n − c 1 log n. Modify the process once more picking k from Binomial(s, d ) instead of
n
from Binomial(m, d ). Let n p′ be the probability that cc(v) is small for the modified process.
Clearly, p ′ ≥ p, so it suffices to prove that p ′ is a constant strictly less than one. The
mean of the binomial now is d 1 = sd/n which is strictly greater than one. It is clear that
the probability that the modified process ends before c 1 log n vertices are discovered is at
least the probability for the original process, since picking from n − c 1 log n vertices has
decreased the number of newly discovered vertices each time. Modifying the process so
that the newly discovered vertices are picked from a fixed size set, converts the problem to
what is called a branching process. Branching processes are discussed further in Section
4.4, but we describe here everything that is needed for our analysis of the giant component.
A branching process is a method for creating a possibly infinite random tree. There
is a nonnegative integer-valued random variable y that is the number of children of the
node being explored. First, the root v of the tree chooses a value of y according to the
distribution of y and spawns that number of children. Each of the children independently
chooses a value according to the same distribution of y and spawns that many children.
The process terminates when all of the vertices have spawned children. The process may
go on forever. If it does terminate with a finite tree, we say that the process has become
“extinct”. Let Binomial(s, d ) be the distribution of y. Let q be the probability of extinction.
Then, q ≥ p ′ , since, the breadth first search terminating with at most c 1 log n
n
vertices is one way of becoming extinct. Let p i = ( s
i)
(d/n) i (1 − (d/n)) s−i be the probability
that y spawns i children. We have ∑ s
i=0 p i = 1 and ∑ s
i=1 ip i = E(y) = ds/n > 1.
The depth of a tree is at most the number of nodes in the tree. Let a t be the probability
that the branching process terminates at depth at most t. If the root v has no children,
then the process terminates with depth one where the root is counted as a depth one node
which is at most t. If v has i children, the process from v terminates at depth at most t if
and only if the i sub processes, one rooted at each child of v terminate at depth t − 1 or
less. The i processes are independent, so the probability that they all terminate at depth
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