Two days ago it occurred to me that almost all simple graphs are small world networks in the sense that if $G_N$ is a simple graph with $N$ nodes sampled from the Erdös-Rényi random graph distribution with probability half then when $N$ is large:
\begin{equation} \mathbb{E}[d(v_i,v_j)] \leq \log_2 N \tag{1} \end{equation}
My strategy for proving this was to show that when $N$ is large, $\forall v_i \in G_N$ there exists a chain of distinct nodes of length $\log_2 N$ originating from $v_i$ almost surely. This implies that:
\begin{equation} \forall v_i, v_j \in G_N, d(v_i,v_j) \leq \log_2 N \tag{2} \end{equation}
almost surely when $N$ is large.
Now, by using the above method of proof I managed to show that almost all simple graphs are very small in the sense that:
\begin{equation} \mathbb{E}[d(v_i,v_j)] \leq \log_2\log_2 N \tag{3} \end{equation}
when $N$ tends to infinity. My question is whether there is an elementary proof that almost all simple graphs are very small world networks and if so in what sense are small world networks special?
Note: I would consider the probabilistic proof I found elementary though I am not a graph theorist so I am not sure whether it's simpler or more complex than the standard proof.