Phase transition in a power-law uniform hypergraph

TL;DR

This paper introduces a power-law uniform hypergraph model where hyperedge formation depends on vertex weights, revealing phase transitions in hyperedge count at alpha=1 and in loose 2-cycle count at alpha=1 and 2.

Contribution

It models hypergraphs with power-law distributed vertex weights and characterizes phase transitions in hyperedge and 2-cycle counts based on the exponent alpha.

Findings

Phase transition in hyperedge count at alpha=1.

Phase transition in loose 2-cycle count at alpha=1 and 2.

Hypergraph properties depend critically on the power-law exponent.

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 $\alpha=2$.