Phase transitions in a power-law uniform hypergraph

We propose a power-law m-uniform random hypergraph on n vertices. In this hypergraph, each vertex is independently assigned a random weight from a power-law distribution with exponent alpha is an element of(0,infinity). The hyperedge probabilities are defined as functions of the random weights. We characterize the number of hyperedges and the number of loose 2-cycles. There is a phase transition phenomenon for the number of hyperedges at alpha = 1. Interestingly, for the number of loose 2-cycles, phase transitions occur at both alpha = 1 and alpha = 2.