# A Modern Proof of Erdos and Renyi's 1959 Random Graph Paper?

In their paper, Erdos and Renyi consider a random graph with a fixed number of edges, as opposed to the more modern approach of adding each edge independently with probability $$p$$. From what I understand (both intuitively and from searching around) their results on connectivity still hold if one use the more standard $$G(n,p)$$ model. I have attempted to generalize their proof ideas to this setting, but was unsuccessful. I was hoping that someone could either sketch or refer me to a proof of their connectivity results in the $$G(n,p)$$ setting. Ideally this proof would be in the same spirit as the original proof, but I'd accept any paper that works.

One can also deduce the $$G(n,p)$$ result from the $$G(n,m)$$ result: one way to generate $$G(n,p)$$ is to choose $$m$$ from the binomial distribution with $$\binom{n}{2}$$ trials and probability $$p$$, then generate $$G(n,m)$$. Since the standard deviation of $$m$$ (in the range of $$p$$ you care about) is around $$\sqrt{n}$$, and the threshold window for connectivity is much larger (corresponds to roughly $$n$$ edges) you don't lose anything by this conversion. Or to put it another way: the values of $$m$$ you get with high probability give models $$G(n,m)$$ which all have essentially the same probability of being connected.