On the tree-depth and tree-width in heterogeneous random graphs

In this note, we investigate the tree-depth and tree-width in a heterogeneous random graph obtained by including each edge $e_{ij}$ $(i\neq j)$ of complete $K_{n}$ over $n$ vertices independently with probability $p_{n}(e_{ij})$. When sequence probabilities satisfies some density assumptions, show both are linear size high probability. Moreover, extend method to weighted graphs non-identical weights capture conditions under which is bounded constant.