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 a giant component in the network. Specifically, when the degrees of a node for different interactions in a duplex Erdős-Rényi network are maximally correlated, the network contains the giant component for any nonzero link density. In contrast, when the degrees of a node are maximally anti-correlated, the emergence of the 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.

Distinct channels of interaction in a complex networked system define network layers, which coexist and cooperate for the system's function. Towards understanding such multiplex systems, we propose a modeling framework based on coevolution of network layers, with a class of minimalistic growing network models as working examples. We examine how the entangled growth of coevolving layers can shape the network structure and show analytically and numerically that the coevolution can induce strong degree correlations across layers, as well as modulate degree distributions. We further show that such a coevolution-induced correlated multiplexity can alter the system's response to the dynamical process, exemplified by the suppressed susceptibility to a social cascade process.

Typical complex system operates through multiple types of interactions between its constituents. The collective function of these multiple interactions, or multiple network layers, is often non-additive, resulting in nontrivial effects on the network structure and dynamics. To better model such situations, the concept of multiplex network, the network with explicit multiple types of links, has recently been applied. In this contribution, we survey recent studies on this subject, focused on the notion of correlated multiplexity. Empirical multiplex network analysis as well as analytical results on the random graph models of correlated multiplex networks are presented, followed by a brief summary of dynamical processes on multiplex networks. It is illustrated that a multiplex complex system can indeed exhibit structural and dynamical properties that cannot be represented by its individual layer's properties alone, establishing the network multiplexity as an essential ingredient in the new physics of "network of networks."

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