Weighted geometric mean of large-scale matrices: numerical analysis and algorithms

Computing the weighted geometric mean of large sparse matrices is an operation that tends to become rapidly intractable, when the size of the matrices involved grows. However, if we are not interested in the computation of the matrix function itself, but just in that of its product times a vector, the problem turns simpler and there is a chance to solve it even when the matrix mean would actually be impossible to compute. Our interest is motivated by the fact that this calculation has some practical applications, related to the preconditioning of some operators arising in domain decomposition of elliptic problems. In this thesis, we explore how such a computation can be efficiently performed. First, we exploit the properties of the weighted geometric mean and find several equivalent ways to express it through real powers of a matrix. Hence, we focus our attention on matrix powers and examine how well-known techniques can be adapted to the solution of the problem at hand. In particular, we consider two broad families of approaches for the computation of f(A) v, namely quadrature formulae and Krylov subspace methods, and generalize them to the pencil case f(A\B) v. Finally, we provide an extensive experimental evaluation of the proposed algorithms and also try to assess how convergence speed and execution time are influenced by some characteristics of the input matrices. Our results suggest that a few elements have some bearing on the performance and that, although there is no best choice in general, knowing the conditioning and the sparsity of the arguments beforehand can considerably help in choosing the best strategy to tackle the problem.

The Matrix Method of Determining the Fault Tolerance Degree of a Computer Network Topology

This work presents a theoretical-graph method of determining the fault tolerance degree of the computer network interconnections and nodes. Experimental results received from simulations of this method over a distributed computing network environment are also presented.

Bethe graphs attached to the vertices of a connected graph: a spectral approach

A weighted Bethe graph $B$ is obtained from a weighted generalized Bethe tree by identifying each set of children with the vertices of a graph belonging to a family $F$ of graphs. The operation of identifying the root vertex of each of $r$ weighted Bethe graphs to the vertices of a connected graph $\mathcal{R}$ of order $r$ is introduced as the $\mathcal{R}$-concatenation of a family of $r$ weighted Bethe graphs. It is shown that the Laplacian eigenvalues (when $F$ has arbitrary graphs) as well as the signless Laplacian and adjacency eigenvalues (when the graphs in $F$ are all regular) of the $\mathcal{R}$-concatenation of a family of weighted Bethe graphs can be computed (in a unified way) using the stable and low computational cost methods available for the determination of the eigenvalues of symmetric tridiagonal matrices. Unlike the previous results already obtained on this topic, the more general context of families of distinct weighted Bethe graphs is herein considered.

Brain network analysis of patients with Temporal Lobe Epilepsy

Patients with Temporal Lobe Epilepsy (TLE) suffer from widespread subtle white matter abnormalities and abnormal functional connectivity extending beyond the affected lobe, as revealed by Diffusion Tensor MR Imaging, volumetric and functional MRI studies. Diffusion Spectrum Imaging (DSI) is a diffusion imaging technique with high angular resolution for improving the mapping of white matter pathways. In this study, we used DSI, connectivity matrices and topological measures to investigate how the alteration in structural connectivity influences whole brain structural networks. Eleven patients with right-sided TLE and hippocampal sclerosis and 18 controls underwent our DSI protocol at 3T. The cortical and subcortical grey matters were parcellated into 86 regions of interest and the connectivity between every region pair was estimated using global tractography and a connectivity matrix (the adjacency matrix of the structural network). We then compared the networks of patients and controls using topological measures. In patients, we found a higher characteristic path length and a lower clustering coefficient compared to controls. Local measures at node level of the clustering and efficiency showed a significant difference after a multiple comparison correction (Bonferroni). These significant nodes were located within as well outside the temporal lobe, and the localisation of most of them was consistent with regions known to be part of epileptic networks in TLE. Our results show altered connectivity patterns that are concordant with the mapping of functional epileptic networks in patients with TLE. Further studies are needed to establish the relevance of these findings for the propagation of epileptic activity, cognitive deficits in medial TLE and outcome of epilepsy surgery in individual patients.

A Note on Energy of Some Graphs

Eigenvalue of a graph is the eigenvalue of its adjacency matrix. The energy of a graph is the sum of the absolute values of its eigenvalues. In this note we obtain analytic expressions for the energy of two classes of regular graphs.

SOME NEW INTEGRAL GRAPHS

The eigenvalue of a graph is the eigenvalue of its adjacency matrix . A graph G is integral if all of its cigenvalues are integers. In this paper some new classes of integral graphs are constructed.

On infinite graphs and related matrices

This thesis Entitled On Infinite graphs and related matrices.ln the last two decades (iraph theory has captured wide attraction as a Mathematical model for any system involving a binary relation. The theory is intimately related to many other branches of Mathematics including Matrix Theory Group theory. Probability. Topology and Combinatorics . and has applications in many other disciplines..Any sort of study on infinite graphs naturally involves an attempt to extend the well known results on the much familiar finite graphs. A graph is completely determined by either its adjacencies or its incidences. A matrix can convey this information completely. This makes a proper labelling of the vertices. edges and any other elements considered, an inevitable process. Many types of labelling of finite graphs as Cordial labelling, Egyptian labelling, Arithmetic labeling and Magical labelling are available in the literature. The number of matrices associated with a finite graph are too many For a study ofthis type to be exhaustive. A large number of theorems have been established by various authors for finite matrices. The extension of these results to infinite matrices associated with infinite graphs is neither obvious nor always possible due to convergence problems. In this thesis our attempt is to obtain theorems of a similar nature on infinite graphs and infinite matrices. We consider the three most commonly used matrices or operators, namely, the adjacency matrix

Optimizing the Dynamic Distribution of Data-stream for High Speed Communications

The performances of high-speed network communications frequently rest with the distribution of data-stream. In this paper, a dynamic data-stream balancing architecture based on link information is introduced and discussed firstly. Then the algorithms for simultaneously acquiring the passing nodes and links of a path between any two source-destination nodes rapidly, as well as a dynamic data-stream distribution planning are proposed. Some related topics such as data fragment disposal, fair service, etc. are further studied and discussed. Besides, the performance and efficiency of proposed algorithms, especially for fair service and convergence, are evaluated through a demonstration with regard to the rate of bandwidth utilization. Hoping the discussion presented here can be helpful to application developers in selecting an effective strategy for planning the distribution of data-stream.

Periodic Reordering

For many networks in nature, science and technology, it is possible to order the nodes so that most links are short-range, connecting near-neighbours, and relatively few long-range links, or shortcuts, are present. Given a network as a set of observed links (interactions), the task of finding an ordering of the nodes that reveals such a range-dependent structure is closely related to some sparse matrix reordering problems arising in scientific computation. The spectral, or Fiedler vector, approach for sparse matrix reordering has successfully been applied to biological data sets, revealing useful structures and subpatterns. In this work we argue that a periodic analogue of the standard reordering task is also highly relevant. Here, rather than encouraging nonzeros only to lie close to the diagonal of a suitably ordered adjacency matrix, we also allow them to inhabit the off-diagonal corners. Indeed, for the classic small-world model of Watts & Strogatz (1998, Collective dynamics of ‘small-world’ networks. Nature, 393, 440–442) this type of periodic structure is inherent. We therefore devise and test a new spectral algorithm for periodic reordering. By generalizing the range-dependent random graph class of Grindrod (2002, Range-dependent random graphs and their application to modeling large small-world proteome datasets. Phys. Rev. E, 66, 066702-1–066702-7) to the periodic case, we can also construct a computable likelihood ratio that suggests whether a given network is inherently linear or periodic. Tests on synthetic data show that the new algorithm can detect periodic structure, even in the presence of noise. Further experiments on real biological data sets then show that some networks are better regarded as periodic than linear. Hence, we find both qualitative (reordered networks plots) and quantitative (likelihood ratios) evidence of periodicity in biological networks.

Centrality and spectral radius in dynamic communication networks

We explore the influence of the choice of attenuation factor on Katz centrality indices for evolving communication networks. For given snapshots of a network observed over a period of time, recently developed communicability indices aim to identify best broadcasters and listeners in the network. In this article, we looked into the sensitivity of communicability indices on the attenuation factor constraint, in relation to spectral radius (the largest eigenvalue) of the network at any point in time and its computation in the case of large networks. We proposed relaxed communicability measures where the spectral radius bound on attenuation factor is relaxed and the adjacency matrix is normalised in order to maintain the convergence of the measure. Using a vitality based measure of both standard and relaxed communicability indices we looked at the ways of establishing the most important individuals for broadcasting and receiving of messages related to community bridging roles. We illustrated our findings with two examples of real-life networks, MIT reality mining data set of daily communications between 106 individuals during one year and UK Twitter mentions network, direct messages on Twitter between 12.4k individuals during one week.

On the radius of centrality in evolving communication networks

In this article, we investigate how the choice of the attenuation factor in an extended version of Katz centrality influences the centrality of the nodes in evolving communication networks. For given snapshots of a network, observed over a period of time, recently developed communicability indices aim to identify the best broadcasters and listeners (receivers) in the network. Here we explore the attenuation factor constraint, in relation to the spectral radius (the largest eigenvalue) of the network at any point in time and its computation in the case of large networks. We compare three different communicability measures: standard, exponential, and relaxed (where the spectral radius bound on the attenuation factor is relaxed and the adjacency matrix is normalised, in order to maintain the convergence of the measure). Furthermore, using a vitality-based measure of both standard and relaxed communicability indices, we look at the ways of establishing the most important individuals for broadcasting and receiving of messages related to community bridging roles. We compare those measures with the scores produced by an iterative version of the PageRank algorithm and illustrate our findings with two examples of real-life evolving networks: the MIT reality mining data set, consisting of daily communications between 106 individuals over the period of one year, a UK Twitter mentions network, constructed from the direct \emph{tweets} between 12.4k individuals during one week, and a subset the Enron email data set.

Designing Novel Antitrypanosomal Agents from a Mixed Graph-Theoretical Substructural Approach

Chagas disease is nowadays the most serious parasitic health problem. This disease is caused by Trypanosoma cruzi. The great number of deaths and the insufficient effectiveness of drugs against this parasite have alarmed the scientific community worldwide. In an attempt to overcome this problem, a model for the design and prediction of new antitrypanosomal agents was obtained. This used a mixed approach, containing simple descriptors based on fragments and topological substructural molecular design descriptors. A data set was made up of 188 compounds, 99 of them characterized an antitrypanosomal activity and 88 compounds that belong to other pharmaceutical categories. The model showed sensitivity, specificity and accuracy values above 85%. Quantitative fragmental contributions were also calculated. Then, and to confirm the quality of the model, 15 structures of molecules tested as antitrypanosomal compounds (that we did not include in this study) were predicted, taking into account the information on the abovementioned calculated fragmental contributions. The model showed an accuracy of 100% which means that the ""in silico"" methodology developed by our team is promising for the rational design of new antitrypanosomal drugs. (C) 2009 Wiley Periodicals, Inc. J Comput Chem 31: 882-894. 2010

Design of novel antituberculosis compounds using graph-theoretical and substructural approaches

The increasing resistance of Mycobacterium tuberculosis to the existing drugs has alarmed the worldwide scientific community. In an attempt to overcome this problem, two models for the design and prediction of new antituberculosis agents were obtained. The first used a mixed approach, containing descriptors based on fragments and the topological substructural molecular design approach (TOPS-MODE) descriptors. The other model used a combination of two-dimensional (2D) and three-dimensional (3D) descriptors. A data set of 167 compounds with great structural variability, 72 of them antituberculosis agents and 95 compounds belonging to other pharmaceutical categories, was analyzed. The first model showed sensitivity, specificity, and accuracy values above 80% and the second one showed values higher than 75% for these statistical indices. Subsequently, 12 structures of imidazoles not included in this study were designed, taking into account the two models. In both cases accuracy was 100%, showing that the methodology in silico developed by us is promising for the rational design of antituberculosis drugs.

Enumeração de espectro de distâncias de esquemas de modulação codificada em treliça empregando codificação turbo

In this work, a performance analysis of transmission schemes employing turbo trellis coded modulation. In general, the performance analysis of such schemes is guided by evaluating the error probability of these schemes. The exact evaluation of this probability is very complex and inefficient from the computational point of view, a widely used alternative is the use of union bound of error probability, because of its easy implementation and computational produce bounds that converge quickly. Since it is the union bound, it should use to expurge some elements of distance spectrum to obtain a tight bound. The main contribution of this work is that the listing proposal is carried out from the puncturing at the level of symbol rather than bit-level as in most works of literature. The main reason for using the symbol level puncturing lies in the fact that the enummerating function of the turbo scheme is obtained directly from complex sequences of signals through the trellis and not indirectly from the binary sequences that require further binary to complex mapping, as proposed by previous works. Thus, algorithms can be applied through matrix from the adjacency matrix, which is obtained by calculating the distances of the complex sequences of the trellis. This work also presents two matrix algorithms for state reduction and the evaluation of the transfer function of this. The results presented in comparisons of the bounds obtained using the proposed technique with some turbo codes of the literature corroborate the proposition of this paper that the expurgated bounds obtained are quite tight and matrix algorithms are easily implemented in any programming software language

Aninhamento em redes bipartidas

We present a nestedness index that measures the nestedness pattern of bipartite networks, a problem that arises in theoretical ecology. Our measure is derived using the sum of distances of the occupied elements in the adjacency matrix of the network. This index quantifies directly the deviation of a given matrix from the nested pattern. In the most simple case the distance of the matrix element ai,j is di,j = i+j, the Manhattan distance. A generic distance is obtained as di,j = (i¬ + j¬)1/¬. The nestedness índex is defined by = 1 − where is the temperature of the matrix. We construct the temperature index using two benchmarks: the distance of the complete nested matrix that corresponds to zero temperature and the distance of the average random matrix that is defined as temperature one. We discuss an important feature of the problem: matrix occupancy. We address this question using a metric index ¬ that adjusts for matrix occupancy