  We prove that the treewidth of an Erd\"{o}s-R\'{e}nyi random graph $\rg{n,
m}$ is, with high probability, greater than $\beta n$ for some constant $\beta
> 0$ if the edge/vertex ratio $\frac{m}{n}$ is greater than 1.073. Our lower
bound $\frac{m}{n} > 1.073$ improves the only previously-known lower bound. We
also study the treewidth of random graphs under two other random models for
large-scale complex networks. In particular, our result on the treewidth of
\rigs strengths a previous observation on the average-case behavior of the
\textit{gate matrix layout} problem. For scale-free random graphs based on the
Barab\'{a}si-Albert preferential-attachment model, our result shows that if
more than 12 vertices are attached to a new vertex, then the treewidth of the
obtained network is linear in the size of the network with high probability.
