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Abstract
The power method is commonly applied to compute the Perron vector of large adjacency matrices. Blondel et al. [SIAM Rev. 46, 2004] investigated its performance when the adjacency matrix has multiple eigenvalues of the same magnitude. It is well known that the Lanczos method typically requires fewer iterations than the power method to determine eigenvectors with the desired accuracy. However, the Lanczos method demands more computer storage, which may make it impractical to apply to very large problems. The present paper adapts the analysis by Blondel et al. to the Lanczos and restarted Lanczos methods. The restarted methods are found to yield fast convergence and to require less computer storage than the Lanczos method. Computed examples illustrate the theory presented. Applications of the Arnoldi method are also discussed.
Highlights
This paper considers networks that can be represented by simple unweighted graphs, that is, no edge starts and ends at the same node, and there is at most one edge between each pair of distinct nodes
This section discusses the application of the Lanczos and restarted Lanczos methods to the computation of the Perron vector of an undirected connected graph
We report the results obtained for the delicious network for comparison
Summary
This section describes-chained undirected graphs and the structure of their adjacency matrices. These graphs, which are particular bipartite graphs, were introduced in [14] and are defined as follows. The adjacency matrix M ∈ Rn×n , associated with G , has the staircase structure. The structure (5) is the same as (1) It follows from Proposition 1 that the adjacency matrix for an-chained undirected graph has pairs of eigenvalues of the opposite sign, which include the Perron root. The matrix C ∈ R2n×n is defined by PMP T = O It follows that the ± singular values of C are eigenvalues of M. We will discuss the computation of the Perron vector of matrices of the form (6), as well as of matrices of the form (4),
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