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 Math., 8 (2012), pp. 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.


  1. birth and death processes
  2. graph
  3. Langevin equation
  4. mean field
  5. metastability
  6. stochastic simulation algorithm

MSC codes

  1. 91D30
  2. 91C20
  3. 65C20

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Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 1394 - 1410
ISSN (online): 1540-3467


Submitted: 22 November 2021
Accepted: 8 July 2022
Published online: 5 December 2022


  1. birth and death processes
  2. graph
  3. Langevin equation
  4. mean field
  5. metastability
  6. stochastic simulation algorithm

MSC codes

  1. 91D30
  2. 91C20
  3. 65C20



Stefano Di Giovacchino
Department of Engineering and Computer Science and Mathematics, University of L’Aquila, L’Aquila, 67100 Italy.
Desmond J. Higham
School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD United Kingdom.
School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD United Kingdom.

Funding Information

GNCS-INDAM project
Leverhulme Trust: RF/2020-310
The work of the first author was supported by the GNCS-INDAM project and by the PRIN2017-MIUR project 2017JYCLSF “Structure preserving approximation of evolutionary problems.” The work of the second author was supported by the Engineering and Physical Sciences Research Council under grants EP/P020720/1 and EP/V015605/1. The work of the third author was supported by the Leverhulme Trust (RF/2020-310) and the Engineering and Physical Sciences Research Council under grant EP/V006177/1.

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