Abstract: Networks arise in many applications. It is often of interest to be able to
identify the most important nodes of a network or to determine the ease of traveling
between them. We are interested in carrying out these tasks for large undirected and
directed networks. Many quantities of interest can be determined by computing certain
matrix functionals. We discuss how for directed and undirected graphs a few steps of
the Lanczos method in combination with Gauss-type quadrature rules can be applied to
determine estimates of upper and lower bounds for quantities of interest.
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