# Diameter of a random graph

I’m considering the standard Erdös/Renyi $$G(n,p)$$ model where we have $$n$$ nodes and each possible edge is sampled independently with probability $$p = \frac{1}{n^\epsilon}$$.

It is relatively straightforward to show that, starting from any node $$u$$, the expected number of hops to reach every other node is $$1/\epsilon$$. However, this does not say much about the probability of having a diameter of $$1/\epsilon$$. (I could apply Markov’s inequality, but that gives a rather weak bound.)

Thus I’m looking for a reference to a concentration result that states that the diameter of such a random graph is $$O(1/\epsilon)$$ with probability at least $$1 – o(1)$$.