Phase transition in a power-law uniform hypergraph

We propose a power-law m-uniform random hypergraph on n vertexes. In this hypergraph, each vertex is independently assigned a random weight from a power-law distribution with exponent α ∈ (0,∞) and the hyperedge probabilities are defined as functions of the random weights. We characterize the number of hyperedge and the number of loose 2-cycle. There is a phase transition phenomenon for the number of hyperedge at α = 1. Interestingly, for the number of loose 2-cycle, phase transition occurs at both α = 1 and α = 2. These results highlights the significent difference between the proposed random hypergraph and the random Erdös-Rényi hypergraph.

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