Phase transition in a power-law uniform hypergraph

Abstract

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 $\alpha \in (0,\infty )$ 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 $\alpha =1$. Interestingly, for the number of loose 2-cycle, phase transition occurs at both $\alpha =1$ and $\alpha =2$.

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