Two-mode networks exhibiting data loss

Abstract

Motivated by a question arising in the analysis of social networks, we investigate pairs of |$(0,1)$| matrices |$A, B$| such that |$AA^\top =BB^\top$| and |$A^\top A=B^\top B$|⁠. Using the techniques of combinatorial matrix theory, we show how the problem can be analysed in terms of certain linear systems. We construct two large infinite families of pairs of such matrices. One family has an amusing connection with regular tournament matrices, while the other is connected with a generalization of Ryser’s notion of an interchange for a |$(0,1)$| matrix. Not surprisingly, both families of matrices are highly structured.