Let G(N,p) be an Erdos-Renyi graph, where N is the number of vertices, and p is the probability that two distinct vertices form an edge. If I plot 1-b0/N over log(p), then I obtain a curve which looks like a logistic function, where b0 is the number of connected components of G(N,p), and p is in (0,1). If p is not too close to zero, then a logistic function has a very good fit. If this were the true model, then the expected value for b0 would be

E(b0) = N/(1+(pN)^k)

with k = k(N) in (0,1), and at least for p not too close to 0. How can one prove this observation? And what can be said about k(N)? One consequence would be that at the percolation point p = 1/N, one has

E(b0) = N/2.

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