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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 lowerbound $\frac{m}{n} > 1.073$ improves the only previously-known lower bound. Wealso study the treewidth of random graphs under two other random models forlarge-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 theBarab\'{a}si-Albert preferential-attachment model, our result shows that ifmore than 12 vertices are attached to a new vertex, then the treewidth of theobtained network is linear in the size of the network with high probability.

Treewidth of Erd\"{o}s-R\'{e}nyi Random Graphs, Random Intersection Graphs, and Scale-Free Random Graphs