Realizability of hypergraphs and high-dimensional contingency tables with random degrees and marginals

TL;DR

This paper demonstrates that random degree sequences and marginals in high-dimensional hypergraphs and contingency tables can be realized with high probability, contrasting with classical graph cases, and confirms two conjectures of Pak and Panova.

Contribution

It establishes probabilistic realizability results for high-dimensional hypergraphs and contingency tables, providing new insights into their structure and confirming conjectures.

Findings

Random integer partitions are realizable as degree sequences of 3-uniform hypergraphs with high probability.

High-dimensional binary contingency tables with random marginals can be constructed with high probability.

Pyramid-shaped contingency tables cannot be constructed with high probability under random marginals.

Abstract

A result of Deza, Levin, Meesum, and Onn shows that the problem of deciding if a given sequence is the degree sequence of a 3-uniform hypergraph is NP complete. We tackle this problem in the random case and show that a random integer partition can be realized as the degree sequence of a $3$-uniform hypergraph with high probability. These results are in stark contrast with the case of graphs, where a classical result of Erd\H{o}s and Gallai provides an efficient algorithm for checking if a sequence is a degree sequence of a graph and a result of Pittel shows that with high probability a random partition is not the degree sequence of a graph. By the same method, we address analogous realizability problems about high-dimensional binary contingency tables. We prove that if $(\lambda,\mu,\nu)$ are three independent random partitions then with high probability one can construct a three-dimensional binary contingency table with marginals $(\lambda,\mu,\nu)$. Conversely, if one insists that the contingency table forms a pyramid shape, then we show that with high probability one cannot construct such a contingency table. These two results confirm two conjectures of Pak and Panova.