Typical rank of coin-toss power-law random matrices over GF(2)

TL;DR

This paper derives a formula for the typical rank of power-law distributed random matrices over GF(2), revealing phase transitions and behaviors relevant to computational and network problems.

Contribution

It introduces a new ensemble of random matrices with power-law column-sums and provides an explicit formula for their typical rank in large sizes.

Findings

Derived a formula for typical rank based on power-law exponent and matrix shape

Characterized phase diagrams showing rank behavior variations

Identified phase transitions in matrix rank properties

Abstract

Random linear systems over the Galois Field modulo 2 have an interest in connection with problems ranging from computational optimization to complex networks. They are often approached using random matrices with Poisson-distributed or finite column/row-sums. This technical note considers the typical rank of random matrices belonging to a specific ensemble wich has genuinely power-law distributed column-sums. For this ensemble, we find a formula for calculating the typical rank in the limit of large matrices as a function of the power-law exponent and the shape of the matrix, and characterize its behavior through "phase diagrams" with varying model parameters.