Expected size of largest connected component

Question

I have a sample space of all the undirected graphs having n vertices and m edges. Then what is the expected value of the size of the largest connected component?

My original problem is that using the G(n, p) model (family of graphs having n vertices and any two vertices would be joined by an edge with probability p. See: Erdős–Rényi model), I need to calculate the expected size of the largest connected component in G(n, p). If the above problem can be solved then this original problem can be easily solved.

PS: I need to exactly calculate the value of the expectation for any fixed n and p given in the input and not just upper or lower bounds.

To be blunt, computing an exact value seems hopeless. Compare it to the problem of finding the exact probability of the graph being connected, which is much easier and already has a complicated recursive formula. — Misha Lavrov, Mar 20 '21 at 18:18
@MishaLavrov Yeah I know that would be much computationally intensive, but I only need that for small values of n. — thestyx, Mar 21 '21 at 20:02

score 1 · Answer 1 · answered Mar 21 '21 at 20:48

I am answering my own question as I have solved the original problem.
The solution is extremely computationally intensive.

Let S be our required expected size of the largest connected component in G(n, p). and let $P_i^n$ be the probability that a graph of n vertices formed from G(n, p) have the largest connected component(s) of size $i$. then

$$S = \sum_{i = 1}^{i = n}iP_i^n$$

Now, for the calculation for $P_i^n$, first choose a set of $i$ vertices and define the below terms

$f_i$ = the probability that a specific set of already chosen $i$ vertices is connected
$g_i^n$ = the probability that no vertex from the set of $i$ chosen vertices make an edge to other remaining $(n-i)$ vertices.

Then $P_i^n$ would be $$P_i^n = \binom{n}{i} f_i g_i^n \sum_{j = 1}^{j = i} P_j^{n - i} $$

This gives a recursive relation for $P_i^n$ (Note that in the summation term would go for $j = i$ and not for $j = i - 1$ as there can be more than one connected component of size i can be possible.

Calculation for $g_i^n$
$g_i$ can be easily calculated as the probability of not forming an edge between two vertices is $1 - p$ and there are total $i * (n - i)$ possible edges which shouldn't be formed, thus $$g_i^n = (1 - p)^{i(n - i)}$$

Calculation for $f_i$
The calculation of $f_i$ is described here: Exact probability of random graph being connected
$$f_i = 1 - \sum_{k = 1}^{i - 1}f_k \binom{i - 1}{k - 1} (1 - p)^{k(i - k)}$$

The calculation of $f_i$ is also recursive. Overall the exact calculation for the required expectation $S$ is much computationally intensive.

Practically, generating a random set of graphs from the G(n, p) model, and calculating the average size of the largest connected component using DFS/BFS would be much easier than the exact method.

Expected size of largest connected component

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