  We analyze the connectivity of an $M$-layer network over a common set of
nodes that are active only in a fraction of the layers. Each layer is assumed
to be a subgraph (of an underlying connectivity graph $G$) induced by each node
being active in any given layer with probability $q$. The $M$-layer network is
formed by aggregating the edges over all $M$ layers. We show that when $q$
exceeds a threshold $q_c(M)$, a giant connected component appears in the
$M$-layer network---thereby enabling far-away users to connect using `bridge'
nodes that are active in multiple network layers---even though the individual
layers may only have small disconnected islands of connectivity. We show that
$q_c(M) \lesssim \sqrt{-\ln(1-p_c)}\,/{\sqrt{M}}$, where $p_c$ is the bond
percolation threshold of $G$, and $q_c(1) \equiv q_c$ is its site percolation
threshold. We find $q_c(M)$ exactly for when $G$ is a large random network with
an arbitrary node-degree distribution. We find $q_c(M)$ numerically for various
regular lattices, and find an exact lower bound for the kagome lattice.
Finally, we find an intriguingly close connection between this multilayer
percolation model and the well-studied problem of site-bond percolation, in the
sense that both models provide a smooth transition between the traditional site
and bond percolation models. Using this connection, we translate known
analytical approximations of the site-bond critical region, which are functions
only of $p_c$ and $q_c$ of the respective lattice, to excellent general
approximations of the multilayer connectivity threshold $q_c(M)$.
