Francisco Pedroche

Additive Schwarz Iterations for Markov Chains

A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations of the form

Subdirect sums of nonsingular M-matrices and of their inverses

Ax=b

A simple generalization of Geršgorin's theorem

. The theory applies to singular

SUBDIRECT SUMS OF S-STRICTLY DIAGONALLY DOMINANT MATRICES ∗

M

A new method for comparing rankings through complex networks: Model and analysis of competitiveness of major European soccer leagues

-matrices with one-dimensional null space and is applicable in particular to systems representing ergodic Markov chains, and to certain discretizations of partial differential equations. Additive Schwarz can be seen as a generalization of block Jacobi, where the set of indices defining the diagonal blocks have nonempty intersection; this is called the overlap. The presence of overlap is known to accelerate the convergence of the methods in the nonsingular case. By providing convergence results, as well as some characteristics of the induced splitting, we hope to encourage the use of this additional computational tool for the solution of Markov chains and other singular systems. We present several numerical examples showing that additive Schwarz performs better than block Jacobi. For completeness, a few numerical experiments with block Gauss--Seidel and multiplicative Schwarz are also included.

Leadership groups on Social Network Sites based on personalized PageRank

The question of when the subdirect sum of two nonsingular M -matrices is a nonsingular M -matrix is studied. Su cient conditions are given. The case of inverses of M -matrices is also studied. In particular, it is shown that the subdirect sum of overlapping principal submatrices of a nonsingular M -matrix is a nonsingular M -matrix. Some examples illustrating the conditions presented are also given. AMS subject classi cations. 15A48.

A biplex approach to PageRank centrality: From classic to multiplex networks

It is well known that the spectrum of a given matrix A belongs to the Geršgorin set Γ(A), as well as to the Geršgorin set applied to the transpose of A, Γ(AT). So, the spectrum belongs to their intersection. But, if we first intersect i-th Geršgorin disk Γi(A) with the corresponding disk

Characterization of α1 and α2-matrices

\Gamma_i(A^T)

Sharp estimates for the personalized Multiplex PageRank

, and then we make union of such intersections, which are, in fact, the smaller disks of each pair, what we get is not an eigenvalue localization area. The question is what should be added in order to catch all the eigenvalues, while, of course, staying within the set Γ(A) ∩ Γ(AT). The answer lies in the appropriate characterization of some subclasses of nonsingular H-matrices. In this paper we give two such characterizations, and then we use them to prove localization areas that answer this question.

A MODEL TO CLASSIFY USERS OF SOCIAL NETWORKS BASED ON PAGERANK

Conditions are given which guarantee that the k-subdirect sum of S-strictly diago- nally dominant matrices (S-SDD) is also S-SDD. The same situation is analyzed for SDD matrices. The converse is also studied: given an SDD matrix C with the structure of a k-subdirect sum and positive diagonal entries, it is shown that there are two SDD matrices whose subdirect sum is C.