The Perron-Frobenius theorem has numerous applications, including to Markov chains, economics, and ranking of webpages. In this paper, we develop a class of continuous-time distributed algorithms and a gossip algorithm, which enable each node i in an undirected and connected graph to compute the ith entry of the Perron-Frobenius eigenvector of a symmetric, Metzler, and irreducible matrix associated with the graph, as well as the corresponding eigenvalue, when node i knows only row i of the matrix. We show that each continuous-time distributed algorithm in the class is a nonlinear networked dynamical system with a skew-symmetric structure, whose state is guaranteed to stay on a sphere, remain nonnegative, and converge asymptotically to the said eigenvector. We also show that under a mild assumption on the gossiping pattern, the gossip algorithm is able to do the same.
['Mu Yang', 'Choon Yik Tang']
Distributed computation of the Perron-Frobenius eigenvector