Foundations of Data Science

By Chebyshev inequality
(
)
Prob |x − E(x)| ≥ E(x) ≤ Var(x)
E 2 (x) → 0.
Thus, Prob(x ≤ 0) goes to zero if Var(x) is o (E 2 (x)) .
Corollary 4.4 Let x be a random variable with E(x) > 0. If
then x is almost surely greater than zero.
Proof: If E(x 2 ) ≤ E 2 (x)(1 + o(1)), then
E(x 2 ) ≤ E 2 (x)(1 + o(1)),
V ar(x) = E(x 2 ) − E 2 (x) ≤ E 2 (x)o(1) = o(E 2 (x)).
Second moment arguments are more difficult than first moment arguments since they
deal with variance and without independence we do not have E(xy) = E(x)E(y). In the
triangle example, dependence occurs when two triangles share a common edge. However,
if p = d , there are so few triangles that almost surely no two triangles share a common
n
edge and the lack of statistical independence does not affect the answer. In looking for
a phase transition, almost always the transition in probability of an item being present
occurs when the expected number of items transitions.
Threshold for graph diameter two (two degrees of separation)
We now present the first example of a sharp phase transition for a property. This
means that slightly increasing the edge probability p near the threshold takes us from
almost surely not having the property to almost surely having it. The property is that
of a random graph having diameter less than or equal to two. The diameter of a graph
is the maximum length of the shortest path between a pair of nodes. In other words, the
property is that every pair of nodes has “at most two degrees of separation”.
The following technique for deriving the threshold for a graph having diameter two
is a standard method often used to determine the threshold for many other objects. Let
x be a random variable for the number of objects such as triangles, isolated vertices, or
Hamiltonian circuits, for which we wish to determine a threshold. Then we determine
the value of p, say p 0 , where the expected value of x goes from vanishingly small to unboundedly
large. For p < p 0 almost surely a graph selected at random will not have a
copy of the item. For p > p 0 , a second moment argument is needed to establish that the
items are not concentrated on a vanishingly small fraction of the graphs and that a graph
picked at random will almost surely have a copy.
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