Francisco Pedroche

Universitat Politècnica de València | UPV · Institute for Multidisciplinary Mathematics (im2)

The concept of subdirect sum of matrices (a type of sum of matrices with overlapping blocks) was introduced in 1999 and has been broadly studied. In this paper we extend this concept by introducing the weighted subdirect sum: a sum of matrices with overlapping blocks allowing to give a different weight to the overlapped blocks. This concept natural...

Mathematical analysis of rankings is essential for a wide range of scientific, public, and industrial applications (e.g., group decision-making, organizational methods, R&D sponsorship, recommender systems, voter systems, sports competitions, grant proposals rankings, web searchers, Internet streaming-on-demand media providers, etc.). Recently, som...

In this paper, some results concerning the PageRank versatility measure for multiplex networks are given. This measure extends to the multiplex setting the well‐known classic PageRank. Particularly, we focus on some spectral properties of the Laplacian matrix of the multiplex and on obtaining boundaries for the ranking value of a given node when so...

Measures of centrality in networks defined by means of matrix algebra, like PageRank-type centralities, have been used for over 70 years. Recently, new extensions of PageRank have been formulated and may include a personalization (or teleportation) vector. It is accepted that one of the key issues for any centrality measure formulation is to what e...

Networks are useful to describe the structure of many complex systems. Often, 1 understanding these systems implies the analysis of multiple interconnected networks simultaneously, 2 since the system may be modeled by more than one type of interaction. Multiplex networks are 3 structures capable of describing networks in which the same set of nodes...

Usually, the nodes’ interactions in many complex networks need a more accurate mapping than simple links. For instance, in social networks, it may be possible to consider different relationships between people. This implies the use of different layers where the nodes are preserved and the relationships are diverse, that is, multiplex networks or bi...

Identifying the influential nodes in complex networks is a fundamental and practical topic at the moment. In this paper, a new centrality measure for complex networks is proposed based on two contrasting models that have their common origin in the well-known PageRank centrality. On the one hand, the essence of the model proposed is taken from the A...

In this paper, we present some results about the spectrum of the matrix associated with the computation of the Multiplex PageRank defined by the authors in a previous paper. These results can be considered as a natural extension of the known results about the spectrum of the Google matrix. In particular, we show that the eigenvalues of the transiti...

PageRank can be understood as the stationary distribution of a Markov chain that occurs in a two-layer network with the same set of nodes in both layers: the physical layer and the teleportation layer. In this paper we present some bounds for the extension of this two-layer approach to Multiplex networks, establishing sharp estimates for this Multi...

In this paper we present some theoretical results about the irreducibility of the Laplacian matrix ordered by the Reverse Cuthill-McKee (RCM) algorithm. We consider undirected graphs with no loops consisting of some connected components. RCM is a well-known scheme for numbering the nodes of a network in such a way that the corresponding adjacency m...

In this paper, we present a new view of the PageRank algorithm inspired by multiplex networks. This new approach allows to introduce a new centrality measure for classic complex networks and a new proposal to extend the usual PageRank algorithm to multiplex networks. We give some analytical relations between these new approaches and the classic Pag...

In this paper we analyze families of rankings by studying structural properties of graphs. Given a finite number of elements and a set of rankings of those elements, two elements compete when they exchange their relative positions in at least two rankings, and we can associate an undirected graph to a set of rankings by connecting elements that com...

In this paper we extend the concept of Competitivity Graph to compare series of rankings with ties ({\em partial rankings}). We extend the usual method used to compute Kendall's coefficient for two partial rankings to the concept of evolutive Kendall's coefficient for a series of partial rankings. The theoretical framework consists of a four-layer...

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(A T ). So, the spectrum belongs to their intersection. But, if we first intersect i-th Geršgorin disk Γ i (A) with the corresponding disk , and then we make union of such intersections, which ar...

In this paper, we show a new technique to analyze families of rankings. In particular, we focus on sports rankings and, more precisely, on soccer leagues. We consider that two teams compete when they change their relative positions in consecutive rankings. This allows to define a graph by linking teams that compete. We show how to use some structur...

In this paper we show a new technique to analyze families of rankings. Given a finite number of elements and a family of rankings of those elements, we consider that two elements compete when they change their relative positions in at least two of those rankings. This allows to define an undirected graph, the so-called competitivity graph, by linki...

In this note we give sufficient conditions for the convergence of the iterative algorithm called weighted-average consensus in directed graphs. We study the discrete-time form of this algorithm. We use standard techniques from matrix theory to prove the main result. As a particular case one can obtain well-known results for non-weighted average con...

In this paper we present a new framework to identify leaders in a Social Network Site using the Personalized PageRank vector. The methodology is based on the concept of Leadership group recently introduced by one of the authors. We show how to analyze the structure of the Leadership group as a function of a single parameter. Zachary’s network and a...

We present a model to classify users of Social Networks. In particular, we focus on Social Network Sites. The model is based on the PageRank algorithm. We use the personalization vector to bias the PageRank to some users. We give an explicit expression of the personalization vector that allows the introduction of some typical features of the users...

In this paper we consider undirected graphs with no loops and multiple edges, consisting of k connected components. In these cases, it is well known that one can find a numbering of the vertices such that the adjacency matrix A is block diagonal with k blocks. This also holds for the (unnormalized) Laplacian matrix L= D-A, with D a diagonal matrix...

In this paper new results on personalized PageRank are shown. We consider directed graphs that may contain dangling nodes. The main result presented gives an analytical characterization of all the possible values of the personalized PageRank for any node.We use this result to give a theoretical justification of a recent model that uses the personal...

Por fin un libro que conecta las matemáticas con la realidad de la edificación. En él, numerosos ejemplos redactados por especialistas permiten al lector hacer el seguimiento desde la intuición del problema real hasta la formalización y la solución matemática para permitir el aprendizaje de toda la potencia de cálculo y adquirir precisión frente a...

It is well known that the spectrum of a given matrix A belongs to the Geršgorin set (A), as well as to the Geršgorin set applied to the transpose of A, (A T). So, the spectrum belongs to their intersection. But, if we first intersect i-th Geršgorin disk i (A) with the corresponding disk i (A T), and then we make union of such intersections, which a...

In this paper a new method to classify the users of an SNS (Social Network Site) into groups is shown. The method is based on the PageRank algorithm. Competitivity groups are sets of nodes that compete among each other to gain PageRank via the personalization vector. Specific features of the SNSs (such as number of friends or activity of the users)...

This paper deals with some properties of α1-matrices and α2-matrices which are subclasses of nonsingular H-matrices. In particular, new characterizations of these two subclasses are given, and then used for proving algebraic properties related to subdirect sums and Hadamard products.

In this article, a generalization of a known result about the subdirect sum of two S-SDD (strictly diagonally dominant) matrices is obtained for Σ-SDD matrices. The class of Σ-SDD matrices is a generalization of S-SDD matrices, and it is also a subclass of H-matrices. More precisely, the question of when the subdirect sum and, consequently, the usu...

In this paper we analyze the Competitivity groups in an SNS (Social Network Site). Competitivity groups are sets of nodes that compete among each other to gain Page-Rank via the personalization vector. A new parameter called amplification factor is introduced. Using a test case, the effect of the amplification factor into the number and the structu...

In this communication some new parameters to discern the most impor- tant people in a Social Network Site are presented. These parameters are based on the connectivity of the nodes of the network as well as the dynamic behavior of each node. The parameters take into account some information on the profile of the user. Some numerical examples are pr...

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 whos...

The question of when the subdirect sum of two nonsingular M-matrices is a non-singular M-matrix is studied. Sufficient 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 ill...

A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations of the form Ax = b. The theory applies to singular M -matrices with one-dimensional null space and is applicable in particular to systems representing ergodic Markov chains, and to certain discretizations of partial different...

In recent years, an algebraic framework was introduced for the analysis of convergence of Schwarz methods for the solution of linear systems of the form Ax = b. Within this frame- work, additive and multiplicative Schwarz were shown to converge when the coefficient mat- rix A is a nonsingular M-matrix, or a symmetric positive definite matrix. In th...

Palabras clave: Sistemas lineales, Métodos iterativos de Schwarz, Métodos por bloques, Solapamiento, M-matriz singular. Resumen. Los métodos iterativos de tipo Schwarz son una extensión natural de los méto-dos clásicos por bloques de Gauss-Seidel y Jacobi aplicados a problemas de discretización de ecuaciones diferenciales donde se ha hecho una divi...

In recent years, an algebraic framework was introduced for the analysis of convergence of Schwarz methods for the solution of linear systems of the form Ax = b. Within this framework, additive and multiplicative Schwarz were shown to converge when the coe#cient matrix A is a nonsingular M-matrix, or a symmetric positive definite matrix. In this pap...

The most of the authors about the topic of University Curricula Design, point out the importance of defining the profesional fields, where the future graduates are going to develop their jobs, in a coherent and up-to-date manner, with the aim of using this information in order to select the contents of the different subjects of the syllabus, which...

This comunication describes some image-compression concepts such as the transmitted energy of the digital signal and the relative error of the compressed image in function of the number and magnitude of the singular values used in the Singular Value Decomposition of the matrix that represents the original picture. Comparisons are made with wavelet-...

The question when subdirect sum of two H-matrices is an H-matrix for the case of S-SDD matrices was treated in the paper by Bru et al. (2006), where some sucient conditions were given. Motivated by the same question, using a dierent technique, we are going to give a simplified proof of the main theorem from the paper above followed by its generaliz...

We denote by H0 the subclass of H-matrices consisting of all the matrices that lay simultaneously on the classes of doubly diagonally dominant (DDD) matrices (A =( aij) ∈ Cn×n : |aii||ajj |≥ k�=i |aik| k�=j |ajk|,i�= j; see (3)) and S-strictly diagonally dominant (S-SDD) matrices; see (1), (2). Notice that strictly doubly diagonally dominant matric...

A convergence analysis is presented for additive Schwarz iterations when applied to consistent singular systems of equations Ax = b. The theory applies to singular M-matrices with one-dimensional null space, and is ap- plicable in particular to systems representing ergodic Markov chains. The results are based on an algebraic formulation of Schwarz...

The fact that the search engine Google is a popular tool known by our students can be used to catch their attention in classroom. Google recognizes that the PageRank™ algorithm is still the basic tool of the engine. In fact, this algorithm has already caught the interest of the researchers in mathematics and it is an active field of investigation....