Tight information theoretic density bounds for sparse crossbar concentrators

A sparse crossbar (n,m)-concentrator is a bipartite graph with n inputs and m outputs, m⩽n, in which there exists a matching between every m inputs and the m outputs. Using information theoretic arguments this paper shows that the density of sparse crossbar (n,m)-concentrators among all the 2^nm sparse crossbars tends to 0 when m/n<1/2 and it tends to 1 when m/n>1/2