Generalized algebraic connectivity for asymmetric networks

Abstract

The problem of connectivity assessment in an asymmetric network represented by a weighted directed graph is investigated in this article. A power iteration algorithm in a centralized implementation is developed first to compute the generalized algebraic connectivity of asymmetric networks. After properly transforming the Laplacian matrix of the network, two sequences of one-dimensional and two-dimensional subspaces are generated iteratively, one of which converges to the desired subspace spanned by the eigenvector(s) associated with the eigenvalue(s) representing the network's generalized algebraic connectivity. A distributed implementation of the proposed power iteration algorithm is then developed to compute the generalized algebraic connectivity from the viewpoint of each node, which is scalable to any asymmetric network of any size with a fixed message length per node. The convergence analysis of these algorithms is subsequently provided under some weak assumptions. The efficiency of the developed algorithms in computing the network connectivity is then demonstrated by simulations.

On digraphs with polygonal restricted numerical range

“…is called the generalized algebraic connectivity of Γ and it is shown that this value reflects the expected asymptotic convergence rate of cooperative consensus-based algorithms in an asymmetric network represented by Γ. Hence, our investigation of polygonal digraphs will help identify networks for which the algebraic connectivity α(Γ), the generalized algebraic connectivity from [1], and the synchronization parameter µ(L) from [15] are equal.…”