Convergence in inhomogeneous consensus processes with positive diagonals

Abstract

We present a results about convergence of products of row-stochastic matrices which are infinite to the left and all have positive diagonals. This is regarded as in inhomogeneous consensus process where confidence weights may change in every time step but where each agent has a little bit of self confidence. The positive diagonal leads to a fixed zero pattern in certain subproducts of the infinite product. We discuss the use of the joint spectral radius on the set of the evolving subproducts and conditions on the subprodutcs to ensure convergence of parts of the infinite product to fixed rank-1-matrices on the diagonal. If the positive minimum of each matrix is uniformly bounded from below the boundedness of the length of intercommunication intervals is important to ensure convergence. We present a small improvement. A slow increase as quick as log(log(t)) in the length of intercommunication intervals is acceptable.