Rank of the vertex-edge incidence matrix of $r$-out hypergraphs

TL;DR

This paper analyzes the rank properties of sparse Boolean matrices derived from 1-out 3-uniform hypergraphs, revealing a positive probability of invertibility and contrasting with other hypergraph models.

Contribution

It introduces the rank distribution of vertex-edge incidence matrices of 1-out hypergraphs, showing a positive constant probability of full rank and finite null space.

Findings

Probability of full rank is about 0.2574 for large n

Null space is finite, contrasting with other hypergraph models

Co-rank distribution varies with hypergraph type and field

Abstract

We consider a space of sparse Boolean matrices of size $n \times n$, which have finite co-rank over $GF(2)$ with high probability. In particular, the probability such a matrix has full rank, and is thus invertible, is a positive constant with value about $0.2574$ for large $n$. The matrices arise as the vertex-edge incidence matrix of 1-out 3-uniform hypergraphs The result that the null space is finite, can be contrasted with results for the usual models of sparse Boolean matrices, based on the vertex-edge incidence matrix of random $k$-uniform hypergraphs. For this latter model, the expected co-rank is linear in the number of vertices $n$, \cite{ACO}, \cite{CFP}. For fields of higher order, the co-rank is typically Poisson distributed.