6 research outputs found
Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices
We derive a priori residual-type bounds for the Arnoldi approximation of a
matrix function and a strategy for setting the iteration accuracies in the
inexact Arnoldi approximation of matrix functions. Such results are based on
the decay behavior of the entries of functions of banded matrices.
Specifically, we will use a priori decay bounds for the entries of functions of
banded non-Hermitian matrices by using Faber polynomial series. Numerical
experiments illustrate the quality of the results
Computation of generalized matrix functions
We develop numerical algorithms for the efficient evaluation of quantities
associated with generalized matrix functions [J. B. Hawkins and A. Ben-Israel,
Linear and Multilinear Algebra 1(2), 1973, pp. 163-171]. Our algorithms are
based on Gaussian quadrature and Golub--Kahan bidiagonalization. Block variants
are also investigated. Numerical experiments are performed to illustrate the
effectiveness and efficiency of our techniques in computing generalized matrix
functions arising in the analysis of networks.Comment: 25 paged, 2 figure
Edge manipulation techniques for complex networks with applications to communicability and triadic closure.
Complex networks are ubiquitous in our everyday life and can be used to model a wide variety of phenomena. For this reason, they have captured the interest of researchers from a wide variety of fields. In this work, we describe how to tackle two problems that have their focus on the edges of networks.
Our first goal is to develop mathematically inferred, efficient methods based on some newly introduced edge centrality measures for the manipulation of links in a network. We want to make a small number of changes to the edges in order to tune its overall ability to exchange information according to certain goals. Specifically, we consider the problem of adding a few links in order to increase as much as possible this ability and that of selecting a given number of connections to be removed from the graph in order to penalize it as little as possible. Techniques to tackle these problems are developed for both undirected and directed networks. Concerning the directed case, we further discuss how to approximate certain quantities that are used to measure the importance of edges.
Secondly, we consider the problem of understanding the mechanism underlying triadic closure in networks and we describe how communicability distance functions play a role in this process.
Extensive numerical tests are presented to validate our approaches