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Ising model on clustered networks: A model for opinion dynamics

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  • Baldassarri, Simone
  • Gallo, Anna
  • Jacquier, Vanessa
  • Zocca, Alessandro

Abstract

We study opinion dynamics on networks with a nontrivial community structure, assuming individuals can update their binary opinion as the result of the interactions with an external influence with strength h∈[0,1] and with other individuals in the network. To model such dynamics, we consider the Ising model with an external magnetic field on a family of finite networks with a clustered structure. Assuming a unit strength for the interactions inside each community, we assume that the strength of interaction across different communities is described by a scalar ϵ∈[−1,1], which allows a weaker but possibly antagonistic effect between communities. We are interested in the stochastic evolution of this system described by a Glauber-type dynamics parametrized by the inverse temperature β. We focus on the low-temperature regime β→∞, in which homogeneous opinion patterns prevail and, as such, it takes the network a long time to fully change opinion. We investigate the different metastable and stable states of this opinion dynamics model and how they depend on the values of the parameters ϵ and h. More precisely, using tools from statistical physics, we derive rigorous estimates in probability, expectation, and law for the first hitting time between metastable (or stable) states and (other) stable states, together with tight bounds on the mixing time and spectral gap of the Markov chain describing the network dynamics. Lastly, we provide a full characterization of the critical configurations for the dynamics, i.e., those which are visited with high probability along the transitions of interest.

Suggested Citation

  • Baldassarri, Simone & Gallo, Anna & Jacquier, Vanessa & Zocca, Alessandro, 2023. "Ising model on clustered networks: A model for opinion dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 623(C).
    • Handle: RePEc:eee:phsmap:v:623:y:2023:i:c:s0378437123003667
    DOI: 10.1016/j.physa.2023.128811
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    References listed on IDEAS

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    1. An Lu & Chunhua Sun & Yezheng Liu, 2017. "The Impact of Community Structure on the Convergence Time of Opinion Dynamics," Discrete Dynamics in Nature and Society, Hindawi, vol. 2017, pages 1-7, June.
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    4. Nardi, Francesca R. & Zocca, Alessandro, 2019. "Tunneling behavior of Ising and Potts models in the low-temperature regime," Stochastic Processes and their Applications, Elsevier, vol. 129(11), pages 4556-4575.
    5. Galam, Serge, 1997. "Rational group decision making: A random field Ising model at T = 0," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 238(1), pages 66-80.
    6. Baldassarri, Simone & Nardi, Francesca R., 2022. "Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 107-144.
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    8. Bianchi, Alessandra & Gaudillière, Alexandre, 2016. "Metastable states, quasi-stationary distributions and soft measures," Stochastic Processes and their Applications, Elsevier, vol. 126(6), pages 1622-1680.
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