Dynamics of the threshold model on hypergraphs
Xin-Jian Xu1, Shuang He1, Li-Jie Zhang2
1Department of Mathematics, Shanghai University, Shanghai 200444, China.
Chaos (Woodbury, N.Y.)
|March 2, 2022
Summary
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.
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.
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