2010/11/06 by Salvatore Mandrà, Mandrà, Salvatore, Marco Cosentino Lagomarsino +3
Physics and Astronomy · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #cond-mat.dis-nn
paper · pdf · doi:10.48550/arxiv.1011.1563
arxiv created 2010/11/06 · arxiv updated 2010/11/09
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.