$\1,
\v^{(2)}, \v^{(3)}\cdots \v^{(N)}$ [@Cit:tutorial]. It is well known that the multiplicity of $\lambda_1$ is the number of connected components in the graph [@Cit:tutorial].

Given the Laplacian matrix $\L$, associated with a graph $\mathcal{G}= (V,E)$, spectral clustering divides $\mathcal{G}$ into two clusters by computing the signs of the $N$ elements of the second eigenvector $\v^{(2)}$, or Fiedler vector. For further details about the computation of two or more clusters see [@Cit:tutorial].

There are many algorithms to compute eigenvectors, such as the Lanczos method or orthogonal iteration [@GolubVanLoan96]. Although some of these methods are distributable, convergence is slow [@GolubVanLoan96] and these algorithms do not consider/take advantage of the fact that the matrix for which the eigenvalues and eigenvectors need to be computed is the adjacency matrix of the underlying graph. In [@Cit:kempe], the authors propose an algorithm to compute the first $k$ largest eigenvectors (associated with the first $k$ eigenvalues with greatest absolute value)[^1] of a symmetric matrix. The algorithm in [@Cit:kempe] emulates the behavior of orthogonal iteration using a decentralized process based on gossip algorithms or deterministic random walks on graphs. This approach can be slow as it converges after $O(\tau\log^2 N)$ iterations [@Cit:kempe] where where $\tau$ is the mixing 