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In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
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Higher-order triadic percolation on random hypergraphs.

Hanlin Sun1, Ginestra Bianconi2,3

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Summary
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This study introduces a new framework for analyzing complex hypergraphs with dynamic connections. It reveals that the main component of these networks can exhibit chaotic behavior, similar to the logistic map.

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Area of Science:

  • Complex systems
  • Network science
  • Theoretical physics

Background:

  • Hypergraphs offer a richer representation of complex systems than traditional graphs.
  • Understanding the dynamics of large-scale networks with time-varying connectivity is crucial.

Purpose of the Study:

  • To develop a theoretical framework combining percolation theory and nonlinear dynamics for studying hypergraphs.
  • To investigate higher-order triadic interactions and their impact on network dynamics.
  • To explore generalizations including hierarchical structures and interdependencies.

Main Methods:

  • Development of a comprehensive theoretical framework integrating percolation theory and nonlinear dynamics.
  • Analysis of higher-order triadic percolation on hypergraphs.
  • Investigation of hierarchical structures and alternative regulation mechanisms (node vs. hyperedge).

Main Results:

  • Demonstrated nontrivial dynamics in the giant component of hypergraphs with higher-order triadic interactions.
  • Showed that the fraction of nodes in the giant component follows a route to chaos, consistent with the logistic map universality class.
  • Highlighted the significant impact of combinatorial complexity in hierarchical structures on critical phenomena and dynamics.

Conclusions:

  • The proposed framework provides novel insights into the complex dynamics of evolving hypergraphs.
  • Findings have implications for understanding real-world networks like climate, biological, and brain networks.
  • The study establishes a foundation for analyzing systems with dynamic and higher-order interactions.