The \emph{core-periphery} network architecture proposed by Avin et al. [ICALP 2014] was shown to support fast computation for many distributed algorithms, while being much sparser than the \emph{congested clique}. For being efficient, the core-periphery architecture is however bounded to satisfy three axioms, among which is the capability of the core to emulate the clique, i.e., to implement the all-to-all communication pattern, in $O(1)$ rounds in the \CONGEST\ model. In this paper, we show that implementing all-to-all communication in $k$ rounds can be done in $n$-node networks with roughly $n^2/k$ edges, and this bound is tight. Hence, sparsifying the core beyond just saving a fraction of the edges requires to relax the constraint on the time to simulate the congested clique. We show that, for $p\gg \sqrt{\log n/n}$, a random graph in ${\cal G}_{n,p}$ can, w.h.p., perform the all-to-all communication pattern in $O(\min\{\frac{1}{p^2},n p\})$ rounds. Finally, we show that if the core can emulate the congested clique in $t$ rounds, then there exists a distributed MST construction algorithm performing in $O(t\log n)$ rounds. Hence, for $t=O(1)$, our (deterministic) algorithm improves the best known (randomized) algorithm for constructing MST in core-periphery networks by a factor $\Theta(\log n)$.