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In the evolution of Erdos-Renyi random graph, denoted by $G(n,p)$ with average degree $c=n \cdot p$, I am interested in appearance of other components in supercritical regime ($c>1$).

I know that there is sharp threshold value for $p<\frac{\ln (n)}{n}$ in existence of isolated vertices in $G(n,p)$. We have $c=n \cdot p$, so for fixed value $c$, isolated vertices are appeared when $\ln (n) > c \Longrightarrow n > e^{c}$.

For example, if $c=3$, then for $n > e^{c=3} \simeq 20$ we expect that isolated vertices are appeared in the random graph.

My question is that is there any sharp threshold value for appearing other components in such a graph? In other word, if $n$ grows, then we expect that in addition to the giant component and some isolated vertices, some isolated vertices connect to each other and form $k$-vertices connected components ($k=2,3,\cdots$).

In particular, in my example for fix value $c=3$, we know that if $n>20$, then isolated vertices are appeared. Is there any threshold value $n$ that we can say that other components are appeared?

Thank you in advance

Mohammad Shahraeini
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1 Answers1

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This is not the correct interpretation of the word "threshold" in the context of Erdos-Renyi random graphs.

The claim "$p = \frac{\ln n}{n}$ is the threshold for connectivity in $G(n,p)$" means the following. Pick a function $p = p(n)$. In the limit as $n \to \infty$, the probability that $G(n,p)$ will be connected will:

  • Tend to $1$ if $p \gg \frac{\ln n}{n}$ (in other words, if $\lim_{n \to \infty} \frac{np}{\ln n} = \infty$).
  • Tend to $0$ if $p \ll \frac{\ln n}{n}$ (in other words, if $\lim_{n \to \infty} \frac{np}{\ln n} = 0$).

In your application, $p = \frac 3n$, so this only tells us that for very large $n$, the graph will not be connected but does not make any promises for $n = 20$ in particular. Moreover, the threshold result makes no promises about what happens if $p = \frac{\ln n}{n}$.

For connectivity in particular, the problem is simple enough that we can find an expression for the exact probability that $G(n,p)$ is connected in terms of $n$ and $p$. So we can look at the actual answer to your starting question: what is the probability that $G(n,\frac3n)$ is connected in terms of $n$? Here is a plot of that probability, for $3 \le n \le 30$:

enter image description here

You can see that the probability decreases very smoothly and nothing special happens when $n=20$. (Actually, for deeper reasons, that is about where the probability passes $\frac1e \approx 0.37$, but I don't think that's what you expected, and this does not generalize to other thresholds.)

The probability of not having isolated vertices will have a very similar plot; even for $n$ in this range, having an isolated vertex is by far the likeliest way for a graph not to be connected.

So you see that it does not even make sense to ask a question like "for what $n$ can we say that other components have appeared?" There is not going to be a point at which that probability suddenly jumps from $\approx 0$ to $\approx 1$; there will be a smooth change over many values of $n$.

Misha Lavrov
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