I'm taking a course which follows the book:
High-Dimensional Probability by Roman Vershynin.

There is an exercise (2.4.4) in the book which I have trouble with:
Consider a random graph $G \sim G(n,p)$ with expected degrees $d=o(\log n)$. Show that with high probability (say 0.9), $G$ has a vertex with degree $10d$.

To solve this I want to take a subset of vertices $V' \subset V$ s.t. each two vertices in $V'$ are not neighbors in the graph, so their degrees are independent. Then use Poisson approximation.
But I think that this way the expected degree of the vertices in this group is no longer $d$, and I cant figure out what can I say about $|V'|$?


Self answering for future reference.

We've solved it by taking $V' \subset V$ randomly of size $n^\frac{1}{3}$,
Since $d=o(\log n)$, $p=o(\frac{\log n}{n})$.
The expected number of edges between vertices in $V'$ is bounded by $n^\frac{2}{3}*p=o(\frac{\log n}{n^\frac{1}{3}})$ so with high probability, there are no edges between vertices in $V'$, therefore their degrees are independent.
Now since the degree of each vertex is the sum of Bernoulli Random variables with small probability, we'll use Poisson approximation,

$P(\exists i\leq n \, | \, d'_i = 10d) = 1 - P(\forall i\leq n \, | \, d'_i \neq 10d) = 1 - P(d'_i \neq 10d)^n = 1 - (1 - P(d'_i = 10d))^n$

$P(d'_i = 10d) = \frac{1}{\sqrt{2\pi 10d}}e^{-d}\frac{ed}{10d}^{10d} \geq ... \geq n^{-c}$
(by using $d \geq C_1\log n$)
Which c is a constant that depends on $C_1$, which can be small as we like (since it's small o).

$1 - (1 - P(d'_i = 10d))^n \geq 1 - (1 - n^{-c})^n \geq 1 - e^{-n^{\frac{1}{3}}/n^c} = 1 - e^{-n^{\frac{1}{3}-c}}$
Now we can choose c as we like to get the desired lower bound on the probability.

  • 2
    $\begingroup$ Your answer works fine, but I want to mention that we don't need to find an independent set $V'$. You can choose any $V'$ of reasonable size and simply say that for every $v \in V'$, the number of neighbors $v$ has outside $V'$ is binomial, and these binomial variables are independent. Then argue that some $v \in V'$ has at least $10d$ neighbors outside $V'$, and therefore has degree at least $10d$. $\endgroup$ – Misha Lavrov Mar 8 at 15:25
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    $\begingroup$ The question asks one with exactly 10d, I don't know how much it'll affect your answer, but it is interesting to look at it that way as well! $\endgroup$ – shahaf finder Mar 8 at 17:33
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    $\begingroup$ Exactly $10d$? Huh. Well, we could first look at the degrees inside $V'$, and then that sets a target (which is at most $10d$) for the number of neighbors we want to find outside $V'$ for each vertex. So it still works out. $\endgroup$ – Misha Lavrov Mar 8 at 17:44
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    $\begingroup$ Ok I understand, so it's another option, thanks! $\endgroup$ – shahaf finder Mar 9 at 18:11

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