  Let $S\subseteq V(G)$ and $\pi_{G}(S)$ denote the maximum number $t$ of
edge-disjoint paths $P_{1},P_{2},\ldots,P_{t}$ in a graph $G$ such that
$V(P_{i})\cap V(P_{j})=S$ for any $i,j\in\{1,2,\ldots,t\}$ and $i\neq j$. If
$S=V(G)$, then $\pi_{G}(S)$ is the maximum number of edge-disjoint spanning
paths in $G$. It is proved [Graphs Combin., 37 (2021) 2521-2533] that deciding
whether $\pi_G(S)\geq r$ is NP-complete for a given $S\subseteq V(G)$. For an
integer $r$ with $2\leq r\leq n$, the $r$-path connectivity of a graph $G$ is
defined as $\pi_{r}(G)=$min$\{\pi_{G}(S)|S\subseteq V(G)$ and $|S|=r\}$, which
is a generalization of tree connectivity. In this paper, we study the $3$-path
connectivity of the $k$-dimensional data center network with $n$-port switches
$D_{k,n}$ which has significate role in the cloud computing, and prove that
$\pi_{3}(D_{k,n})=\lfloor\frac{2n+3k}{4}\rfloor$ with $k\geq 1$ and $n\geq 6$.
