Endogenous multiplex networks

Agents are typically connected in a multiplex of interrelated networks (layers). Given an initial asymmetric “seed” layer endowed with a nested split graph topology, we characterize the architecture of connections that bind agents in each layer of a multiplex. We propose a sequential multiplex formation game in which agents are free to choose layers and partners. Adding/deleting links on any layer generates intra-layer externalities within that layer and inter-layer externalities across other layers. The inter-layer externality transmitted from one layer to another is positive (negative) if actions on the two layers are strategic complements (substitutes). The flow of intra-layer and inter-layer externalities produce a limit multiplex in which non-seed layers that are complements (substitutes) of the seed inherit nested split graph topologies that are parallel (flip) of the seed in the sense that high-centrality agents in the seed layer occupy high (low) centrality positions in the non-seed layer. This result is shown to hold across duplexes, triplexes, and a sub-class of higher dimensional multiplexes. A number of applications are also discussed.