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Let $G$ be a graph with $n$ vertices, whose average degree is $k$. What is the probability that between any two vertices, there exists a path of length at most $l$? NOTE: For the above problem the random graphs follow a $G(n, p)$ model, i.e., in a random graph, each edge occurs with a probability $p=k/n$.

Andrew Uzzell
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S Maity
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    Trying to revive the discussion... The problem is equivalent of guarantee that the diameter of $G(n,p)$ is bounded below by $l$. Well, results about $D(G(n,p))$ are already known. if $k<1$ the graph isn't connected $a.a.s$... And are already know the right order of $D(G(n,p))$ if $k$ is large enough... Maybe this can help http://www.iis.ee.ic.ac.uk/~m.draief/file/Home_files/Diameter%20of%20Erdos-Renyi%20graphs.pdf – Rodrigo Ribeiro Dec 04 '14 at 23:01
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    See Chapter 3 of Bollobas' "Random Graphs". The distribution of the subgraphs is Poisson, so 1-Exp[-expected number of extant subgraphs] is the existence probability. – apg Sep 06 '19 at 16:57
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    @apkg, would you be willing to expand your comment into an answer? – Andrew Uzzell Nov 10 '21 at 14:00
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    I will shortly yes – apg Nov 10 '21 at 15:16

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