Generalized Power Iteration with Application to Distributed Connectivity Estimation of Asymmetric Networks

TL;DR

This paper introduces a generalized power iteration method for assessing connectivity in asymmetric networks, providing both centralized and distributed algorithms with proven convergence and scalability demonstrated through simulations.

Contribution

It develops a novel power iteration algorithm for asymmetric network connectivity, including a scalable distributed version with convergence analysis.

Findings

The centralized algorithm accurately computes generalized algebraic connectivity.

The distributed algorithm is scalable and converges under weak assumptions.

Simulations confirm the efficiency of the proposed methods.

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.