Largest component in an Erdos Renyi graph

Question

I need to find a $p$ for an Erdos Renyi graph with $n$ nodes so that the largest component in the graph has a size of at least $0.25n$. With simulation we have come up with an answer that $p = c/n$ and in this case $c$ is around $1.2$. However, we have a hard time finding an easy-to-understand source to complement our answer. Is there any source that explicitly says something about what c should be?

Answer 1

In the random graph with edge probability $p=\frac{c}{n}$ (with $c>1$), the largest component is of size $\left(1-\frac{x}{c}+o(1)\right)n$ with probability tending to one as $n$ tends to infinity. Here $0<x<1$ is the solution of the equation $x e^{-x}=c e^{-c}$. So for your question, you can equate $0.25 = \left(1-\frac{x}{c}\right)$, and then find $x$ and after that find $c$ from the definition of $x$ etc (this will of course be an approximation, so for simulation you should probably take $c$ little bit larger than this solution).

For a proof of this fact see for example Theorem 2.14 in Introduction to random graphs by Frieze and Karonski.