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Theory of percolation on hypergraphs.

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Hypergraph robustness requires hypergraph percolation, not factor graph percolation, for systems where interactions fail if any node is removed. This new method reveals distinct thresholds and limits hyper-resilience in complex networks.

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Area of Science:

  • Complex Systems Science
  • Network Science
  • Statistical Physics

Background:

  • Hypergraphs model higher-order interactions in complex systems, often represented as factor graphs.
  • Existing robustness studies use factor-graph percolation, which is insufficient for systems where hyperedge failure occurs upon node removal.

Purpose of the Study:

  • To introduce and develop hypergraph percolation as the correct framework for analyzing robustness in specific complex systems.
  • To differentiate hypergraph percolation from factor graph percolation and explore its implications.

Main Methods:

  • Developed a message-passing theory for hypergraph percolation.
  • Investigated critical behavior using generating function formalism.
  • Performed Monte Carlo simulations on random and real-world network data.

Main Results:

  • The node percolation threshold for hypergraphs is higher than for factor graphs.
  • Unlike ordinary graphs, node and hyperedge percolation thresholds differ on hypergraphs, with node thresholds being higher.
  • Fat-tailed hyperedge cardinality distributions do not induce hyper-resilience in hypergraphs, unlike in factor graphs.

Conclusions:

  • Hypergraph percolation is essential for accurately assessing the robustness of systems like supply chains and biological networks.
  • The distinct percolation thresholds on hypergraphs have significant implications for network resilience and design.
  • The hyper-resilience phenomenon observed in factor graphs is not replicated in hypergraphs due to differences in their percolation dynamics.