Foundations of Data Science

Assume c < 1. Write x = I 1 + I 2 + · · · + I n where I i is the indicator variable indicating
∑
whether vertex i is an isolated vertex. Then E (x 2 ) = n E (Ii 2 ) + 2 ∑ E (I i I j ). Since I i
i=1 i 1 there almost surely
n
are no isolated vertices, and when c < 1 there almost surely are isolated vertices.
Hamilton circuits
So far in establishing phase transitions in the G(n, p) model for an item such as the
disappearance of isolated vertices, we introduced a random variable x that was the number
of occurrences of the item. We then determined the probability p for which the expected
value of x went from zero to infinity. For values of p for which E(x) → 0, we argued that
with high probability, a graph generated at random had no occurrences of x. For values of
x for which E(x) → ∞, we used the second moment argument to conclude that with high
probability, a graph generated at random had occurrences of x. That is, the occurrences
that forced E(x) to infinity were not all concentrated on a vanishingly small fraction of
the graphs. One might raise the question for the G(n, p) graph model, do there exist
items that are so concentrated on a small fraction of the graphs that the value of p where
E(x) goes from zero to infinity is not the threshold? An example where this happens is
Hamilton circuits.
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