  Triadic closure describes the tendency for new friendships to form between
individuals who already have friends in common. It has been argued
heuristically that the triadic closure effect can lead to bistability in the
formation of large-scale social interaction networks. Here, depending on the
initial state and the transient dynamics, the system may evolve towards either
of two long-time states. In this work, we propose and study a hierarchy of
network evolution models that incorporate triadic closure, building on the work
of Grindrod, Higham, and Parsons [Internet Mathematics, 8, 2012, 402--423]. We
use a chemical kinetics framework, paying careful attention to the reaction
rate scaling with respect to the system size. In a macroscale regime, we show
rigorously that a bimodal steady-state distribution is admitted. This behavior
corresponds to the existence of two distinct stable fixed points in a
deterministic mean-field ODE. The macroscale model is also seen to capture an
apparent metastability property of the microscale system.
  Computational simulations are used to support the analysis.

Social and Information Networks;Numerical Analysis;Dynamical Systems;Numerical Analysis