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Area of Science:

  • Graph theory
  • Statistical inference
  • Combinatorics

Background:

  • Inferring hidden structures in complex networks is a significant challenge.
  • Random k-hypergraphs provide a model for systems with multi-way interactions.

Purpose of the Study:

  • To investigate the phase transitions in recovering a hidden matching within weighted random k-hypergraphs.
  • To compare the recovery behavior for different values of k (hyperedge size).

Main Methods:

  • Analysis of weighted random k-hypergraphs with two distinct weight distributions.
  • Large-graph-size limit analysis to identify algorithmic phase transitions.
  • Study of interpolating cases between k=2 (graphs) and k=3 (3-hypergraphs).

Main Results:

  • A first-order algorithmic phase transition is identified for k>2, separating regimes of complete and partial hidden matching recovery.
  • This contrasts with the continuous transition observed in the k=2 (graph) case.
  • The transition shifts from continuous to first-order as the proportion of 3-hyperedges increases in mixed graphs and hypergraphs.

Conclusions:

  • The size of hyperedges (k) critically influences the nature of hidden structure recovery.
  • Understanding these transitions is crucial for applications in network analysis and statistical inference.