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Dynamics of the threshold model on hypergraphs.

Xin-Jian Xu1, Shuang He1, Li-Jie Zhang2

  • 1Department of Mathematics, Shanghai University, Shanghai 200444, China.

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|March 2, 2022
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Summary
This summary is machine-generated.

This study extends the threshold model to hypergraphs, revealing how higher-order interactions influence contagion. Increasing hyperedge size can destabilize or stabilize systems, while heterogeneity impacts fragility and robustness.

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Area of Science:

  • Network Science
  • Complex Systems
  • Sociophysics

Background:

  • The threshold model is a key framework for understanding contagion on social networks.
  • Previous models primarily focused on pairwise interactions, neglecting higher-order group dynamics.

Purpose of the Study:

  • To investigate the threshold model on hypergraphs, analyzing the impact of group interactions (hyperedges) on contagion.
  • To develop a theoretical framework for understanding cascade conditions and system robustness in hypergraph structures.

Main Methods:

  • Utilized generating function technology to derive cascade conditions and analyze the giant component of vulnerable vertices.
  • Developed a theoretical framework considering hyperedges and hyperdegrees.
  • Verified theoretical findings through simulations of meme spreading on random and empirical hypergraphs.

Main Results:

  • Identified a dual role for hyperedges: increasing size can lead to fragility or robustness depending on average hyperdegree.
  • Heterogeneity in thresholds increases system fragility, while heterogeneity in hyperdegrees/hyperedges enhances robustness.
  • Higher vertex hyperdegree correlates with increased activation probability and speed.

Conclusions:

  • Hypergraph structures introduce complex dynamics to contagion processes not captured by traditional network models.
  • System behavior is sensitive to the interplay between hyperedge size, hyperdegree, and heterogeneity.
  • The findings have implications for understanding information diffusion and system resilience in complex, group-interacting systems.