Let $H,G$ be finite simple graphs. Then the density of $H$ in $G$, denoted $d(H,G)$, is defined as the probability that a randomly chosen $|H|$-tuple of vertices of $G$ induce a graph isomorphic to $H$. Note that $$d(H,G)=\frac{\operatorname{ind}(H,G)}{|G|(|G|-1)\cdots (|G|-|H|+1)},$$ where $\operatorname{ind}(H,G)$ is the number of embeddings of $H$ as induced subgraphs of $G$.

A sequence of graphs $(G_n)_{n=1}^\infty$ is convergent if for every finite simple graph $H$, the sequence $d(H,G_n)$ converges as $n\rightarrow\infty$.


In the following article, http://research.microsoft.com/en-us/um/people/borgs/papers/teststoc.pdf, in Example 5 (Random graphs), the authors assert that the sequence of Erdos-Renyi random graphs $(G(n,p))_{n=1}^\infty$ is convergent with probability $1$ (with $p\in[0,1]$ fixed). They claim that "it is not hard (using high concentration results)...". I have some decent knowledge in graph theory, but I am not so comfortable in probability theory, so I haven't succeeded in proving the assertion.

My approach

What I have tried is the following. Let $H$ be a fixed graph, we denote $G_n=G(n,p)$ for some fixed $p\in[0,1]$. We want to prove that the sequence $d(H,G_n)$ is convergent with probability $1$. It is straightforward to compute the expected value $$\mathbb{E}d(H,G_n)=|\operatorname{Aut}(H)|p^{|E(H)|}(1-p)^{\binom{|H|}{2}-|E(H)|},$$ which is independent of $n$. Now, the idea is to show that the random variable $d(H,G_n)$ is highly concentrated around its expected value (as the authors suggest) -- so that we can use Borel-Cantelli lemma. In particular, if we show that $$\operatorname{Pr}[|d(H,G_n)-\mathbb{E}d(H,G_n)|>t_n]\leq q_n,$$ for some sequences $(t_n)_{n=1}^\infty$ and $(q_n)_{n=1}^\infty$ such that $t_n\rightarrow 0$ and $\sum_{n=1}^\infty q_n<\infty$, then we can apply Borel-Cantelli lemma on the sequence of events $E_1,E_2,\ldots$ where $E_n$ is the event $|d(H,G_n)-\mathbb{E}d(H,G_n)|>t_n$ to conclude that with probability $1$, only finitely many $E_n$ occur. This should directly imply that the sequence $d(H,G_n)$ is convergent, right?

The missing piece is the proof that the random variable $d(H,G_n)$ is highly concentrated around its expected value. I know about Chernoff bounds and Chebychev inequality, but it seems that some more involved bound is needed here.

  • $\begingroup$ Are you sure that Chebyshev won't work? I did some quick calculations for $H$ being a triangle and Chebyshev's inequality was sufficient. $\endgroup$ – D Poole Jul 19 '16 at 12:44
  • $\begingroup$ @DPoole Well, I am not even sure what the variance of $d(H,G_n)$ is. Should that be obvious? Could you elaborate on how you did the quick calculations, please? Thanks for answering. $\endgroup$ – JS_ Jul 19 '16 at 12:48
  • $\begingroup$ For a fixed $H$ and $n=|G|$, the denominator of $d(H, G_n)$ is fixed. So we wish to show that the number of triangles in $G_n$ is concentrated around its mean ${n \choose 3}p^3$. If $X$ is the number of triangles, then $X= \sum_{\sigma} 1_\sigma$, where $\sigma$ runs over all possible triangles. Now $var[X] \leq E[X] + \sum_{|\sigma \cap \tau|=2} E[1_\sigma 1_\tau]$ (simple inequality, look at the definition of variance with the sum. The pairs of triangles that do not share edges are independent and do not add to the sum). This latter sum is $O(n^4)$. Chebyshev's Inequality finishes it off. $\endgroup$ – D Poole Jul 19 '16 at 12:56
  • $\begingroup$ @DPoole Denote $d_n=d(H,G_n)$. Okey, so you showed that the variance of $X$ is $O(n^4)$. Since $X=d_n\binom{n}{3}$ we obtain $var[d_n]=O(n^{-2})$. Now, we have that $s_n=\sqrt{var(d_n)}=O(1/n)$, Chebychev's inequality says $Pr[|d_n-E|\geq k_n s_n]\leq 1/k_n^2$. So we need to find the sequence $(k_n)_{n=1}^\infty$ such that $k_n s_n \rightarrow 0$ and $\sum_{n=1}^\infty 1/k_n^2<\infty$. Since $s_n=O(1/n)$, we have that $k_n=o(n)$ -- we need that to force $k_n s_n\rightarrow 0$. On the other hand, convergence of $\sum 1/k_n^2$ is also limiting. What sequence $k_n$ suffices? Pardon my ignorance $\endgroup$ – JS_ Jul 19 '16 at 13:40
  • $\begingroup$ It is easier to use Chebyshev's Inequality on $X$ (which tends to infinity w.h.p.) rather than $d_n$. Namely, by Chebyshev's Inequality, you would have that with probability tending to 1, $|X-E[X]| < E[X]/\sqrt{n}$. On the event that this occurs, what can you say about $d_n$? $\endgroup$ – D Poole Jul 19 '16 at 14:42

Fix $p \in (0, 1)$ and a graph $H$ with $\nu \geq 2$ vertices and $\mu$ edges.

Let us consider the number, $X$, of induced subgraphs of $H$ into $G(n,p)$. Now $$ X = \sum_{\sigma} 1_\sigma, $$ where $\sigma$ runs over all possible $H$ subgraphs of $K_n$ and $1_\sigma$ is the indicator function for whether the edges of $\sigma$ are present in $G(n,p)$. Note that $E[1_\sigma] = p^\mu (1-p)^{{\nu \choose 2}-\mu}$. Further, $$ E[X] = {n \choose \nu} |Aut(H)|p^\mu (1-p)^{{\nu \choose 2}-\mu}, $$ which is $\Theta\left(n^\nu\right)$.

Note that \begin{align*} \text{var}[X] &= \sum_{\sigma, \tau} \left(E[1_\sigma 1_\tau] - E[1_\sigma] E[1_\tau] \right) \\& \leq \sum_{\sigma} E[1_\sigma] + \sum_{\sigma \neq \tau} \left(E[1_\sigma 1_\tau] - E[1_\sigma] E[1_\tau] \right). \end{align*} For those $\sigma$ and $\tau$ that do not share at least two vertices, this latter sum is zero because $\sigma$ and $\tau$ look at only different independent potential edges. Thus \begin{align*} \text{var}[X] &\leq \sum_{\sigma} E[1_\sigma] + \sum_{\sigma \neq \tau, \atop |V(\sigma)\cap V(\tau)|\geq 2} \left(E[1_\sigma 1_\tau] - E[1_\sigma] E[1_\tau] \right) \\ & \leq E[X] + \sum_{\sigma \neq \tau, \atop |V(\sigma)\cap V(\tau)|\geq 2} E[1_\sigma 1_\tau]. \end{align*} This last sum is definitely $O(n^{\nu+\nu-2}) = O(n^{2\nu-2})$. Thus $$ \text{var}[X] = O(n^\nu) + O(n^{2\nu-2}) = O(n^{2\nu-2}). $$ For any $\epsilon_n$, Chebyshev's Inequality says that $$ P(|X-E[X]| > \epsilon_n E[X]) \leq \frac{\text{var}[X]}{\epsilon_n^2 E[X]^2} = O\left(\frac{1}{\epsilon^2 n^2}\right). $$

Choosing $\epsilon = n^{-1/4}$ would give that $$ \sum_{n=1}^{\infty} \frac{1}{\epsilon^2 n^2} $$ converges so that you can use the Borel-Cantelli Lemma as you wanted.


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