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Unanimity rule on networks.

Renaud Lambiotte1, Stefan Thurner, Rudolf Hanel

  • 1GRAPES, Université de Liège, Sart-Tilman, B-4000 Liège, Belgium.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 13, 2007
PubMed
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A critical number of initially activated nodes is needed for a system to fully activate over time. This model explains innovation, evolution, and opinion dynamics on directed networks using a unanimity rule.

Area of Science:

  • Complex systems
  • Network science
  • Computational social science

Background:

  • Models of spreading phenomena often rely on simplified network structures and update rules.
  • Understanding the conditions for large-scale activation is crucial for various applications, from ecological systems to marketing.

Purpose of the Study:

  • To introduce a novel model for innovation, evolution, and opinion dynamics based on a unanimity rule.
  • To investigate the conditions for system-wide activation in directed networks.
  • To analyze the impact of network topology and initial activation levels on spreading dynamics.

Main Methods:

  • Development of a mathematical model for state transitions in a directed network.
  • Analytical solutions for simple network topologies.

Related Experiment Videos

  • Numerical simulations and analysis for random and small-world networks.
  • Investigation of the critical threshold for initial activation.

    • Main Results:

    • A critical initial density of activated nodes is necessary for global activation in the long-time limit.
    • The model exhibits a phase transition dependent on the initial conditions.
    • The degree distribution of the network significantly influences the spreading dynamics.
    • Analytical results are obtained for specific network structures, with detailed analysis for random and small-world networks.

    Conclusions:

    • The unanimity rule provides a robust mechanism for cascading activation in directed networks.
    • The critical initial condition highlights the importance of early adoption or seeding in spreading processes.
    • The model offers a framework for understanding diverse phenomena, including food-chain dynamics and viral marketing, by considering network structure and activation thresholds.