Endogenous Multiplex Networks

Abstract

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. Specifically, an agent can choose any layer to add one link as well as choose any number of layers to delete existing links. Agents can also make changes to the seed layer. Adding/deleting a link 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.

Keywords: Bonacich centrality, multiplex, nested split graphs, network formation, strategic complementarity and substitutability