  Eigenvectors of data matrices play an important role in many computational
problems, ranging from signal processing to machine learning and control. For
instance, algorithms that compute positions of the nodes of a wireless network
on the basis of pairwise distance measurements require a few leading
eigenvectors of the distances matrix. While eigenvector calculation is a
standard topic in numerical linear algebra, it becomes challenging under severe
communication or computation constraints, or in absence of central scheduling.
In this paper we investigate the possibility of computing the leading
eigenvectors of a large data matrix through gossip algorithms.
  The proposed algorithm amounts to iteratively multiplying a vector by
independent random sparsification of the original matrix and averaging the
resulting normalized vectors. This can be viewed as a generalization of gossip
algorithms for consensus, but the resulting dynamics is significantly more
intricate. Our analysis is based on controlling the convergence to stationarity
of the associated Kesten-Furstenberg Markov chain.
