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Triadic closure describes the tendency for new friendships to form betweenindividuals who already have friends in common. It has been arguedheuristically that the triadic closure effect can lead to bistability in theformation of large-scale social interaction networks. Here, depending on theinitial state and the transient dynamics, the system may evolve towards eitherof two long-time states. In this work, we propose and study a hierarchy ofnetwork evolution models that incorporate triadic closure, building on the workof Grindrod, Higham, and Parsons [Internet Mathematics, 8, 2012, 402--423]. Weuse a chemical kinetics framework, paying careful attention to the reactionrate scaling with respect to the system size. In a macroscale regime, we showrigorously that a bimodal steady-state distribution is admitted. This behaviorcorresponds to the existence of two distinct stable fixed points in adeterministic mean-field ODE. The macroscale model is also seen to capture anapparent metastability property of the microscale system.  Computational simulations are used to support the analysis.

A Hierarchy of Network Models Giving Bistability Under Triadic Closure