Foundations of Data Science

No items
E(x) ≥ 0.1
At least one
occurrence
of item in
10% of the
graphs
For 10% of the
graphs, x ≥ 1
Figure 4.6: If the expected fraction of the number of graphs in which an item occurs did
not go to zero, then E (x), the expected number of items per graph, could not be zero.
Suppose 10% of the graphs had at least one occurrence of the item. Then the expected
number of occurrences per graph must be at least 0.1. Thus, E (x) → 0 implies the
probability that a graph has an occurrence of the item goes to zero. However, the other
direction needs more work. If E (x) is large, a second moment argument is needed to
conclude that the probability that a graph picked at random has an occurrence of the
item is non-negligible, since there could be a large number of occurrences concentrated on
a vanishingly small fraction of all graphs. The second moment argument claims that for
a nonnegative random variable x with E (x) > 0, if Var(x) is o(E 2 (x)) or alternatively if
E (x 2 ) ≤ E 2 (x) (1 + o(1)), then almost surely x > 0.
latter case uses what we call the second moment method. The first and second moment
methods are broadly used. We describe the second moment method in some generality
now.
When the expected value of x(n), the number of occurrences of an item, goes to infinity,
we cannot conclude that a graph picked at random will likely have a copy since
the items may all appear on a vanishingly small fraction of the graphs. We resort to a
technique called the second moment method. It is a simple idea based on Chebyshev’s
inequality.
Theorem 4.3 (Second Moment method) Let x(n) be a random variable with E(x) > 0.
If
( )
Var(x) = o E 2 (x) ,
then x is almost surely greater than zero.
Proof: If E(x) > 0, then for x to be less than or equal to zero, it must differ from its
expected value by at least its expected value. Thus,
(
)
Prob(x ≤ 0) ≤ Prob |x − E(x)| ≥ E(x) .
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