A Hierarchy of Network Models Giving Bistability Under Triadic Closure

TL;DR

This paper introduces a hierarchy of network models incorporating triadic closure, demonstrating how such models can exhibit bistability with two stable long-term states, supported by rigorous analysis and simulations.

Contribution

It develops a new hierarchy of network evolution models with triadic closure, using a chemical kinetics framework to prove bistability and metastability properties.

Findings

Bimodal steady-state distribution in macroscale regime

Existence of two stable fixed points in mean-field ODE

Metastability observed in microscale system

Abstract

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.