  Nodes in a complex networked system often engage in more than one type of
interactions among them; they form a multiplex network with multiple types of
links. In real-world complex systems, a node's degree for one type of links and
that for the other are not randomly distributed but correlated, which we term
correlated multiplexity. In this paper we study a simple model of multiplex
random networks and demonstrate that the correlated multiplexity can
drastically affect the properties of giant component in the network.
Specifically, when the degrees of a node for different interactions in a duplex
Erdos-Renyi network are maximally correlated, the network contains the giant
component for any nonzero link densities. In contrast, when the degrees of a
node are maximally anti-correlated, the emergence of giant component is
significantly delayed, yet the entire network becomes connected into a single
component at a finite link density. We also discuss the mixing patterns and the
cases with imperfect correlated multiplexity.
