  For a connected graph $G$ of order at least $2$ and $S\subseteq V(G)$, the
\emph{Steiner distance} $d_G(S)$ among the vertices of $S$ is the minimum size
among all connected subgraphs whose vertex sets contain $S$. Let $n$ and $k$ be
two integers with $2\leq k\leq n$. Then the \emph{Steiner $k$-eccentricity
$e_k(v)$} of a vertex $v$ of $G$ is defined by $e_k(v)=\max
\{d_G(S)\,|\,S\subseteq V(G), \ |S|=k, \ and \ v\in S\}$. Furthermore, the
\emph{Steiner $k$-diameter} of $G$ is $sdiam_k(G)=\max \{e_k(v)\,|\, v\in
V(G)\}$. In this paper, we investigate the Steiner distance and Steiner
$k$-diameter of Cartesian and lexicographical product graphs. Also, we study
the Steiner $k$-diameter of some networks.
