The symmetric eigenvalue problem: stochastic perturbation theory and some network applications.

This thesis is concerned with stochastic perturbation theory of the symmetric eigen-value problem. In particular, we provide results about the probability of interchanges in the ordering of the eigenvalues and changes in the eigenvectors of symmetric matrices subject to stochastic perturbations. In this analysis we use a novel combination of traditional Numerical Linear Algebra, Perturbation Theory and Probability Theory. The motivation for this study arises from reliability of spectral clustering of networks, when network data is subject to noise. As far as we are aware, there is nothing comparable in the literature. Further, we make conjectures from which we derive an asymptotic relation between the distributions of the largest eigenvalue and the 2-norm of random symmetric ma- trices, whose entries above the main diagonal are independent, identically distributed random variables with probability density functions being symmetric with respect to zero, including matrices from the Gaussian Orthogonal Ensemble (GOE). As far as we know, some of these conjectures are not new (possibly only as conjectures) but we are not aware of any proofs. Also, we consider networks of coupled oscillators. In their analysis we use both, knowledge of dynamical systems and spectral properties of non-negative matrices. As a result, we present an algorithm, which uncovers the \master-slave" structure of the network. With its help, the analysis of the dynamics and the entrainment of the entire network can be reduced to considering only few of the oscillators, those whose dynamics determine the behaviour of the rest. This can be helpful in large networks exhibiting the \master-slave" structure. Finally, we consider similarities of spectral clustering with respect to di®erent matrices which can be associated with a given network. In particular, we compare clustering of products of Path graphs with respect to two di®erent matrices: the Laplacian and the Normalised Laplacian matrices of the graph. We make the comparison by constructing a Homotopy between two eigenvalue problems and, using some Linear Algebra techniques, we show that the two matrices give similar spectral clusterings when applied to products of Path graphs.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Analytical determination of the structure of the outer crust of a cold nonaccreted neutron star: extension to strongly quantizing magnetic fields

The iterative method recently proposed for determining the internal constitution of the outer crust of a nonaccreted neutron star is extended to magnetars by taking into account the Landau-Rabi quantization of electron motion induced by the presence of a very high magnetic field. It is shown that in the strongly quantizing regime, the method can be efficiently implemented using new analytical solutions for the transitions between adjacent crustal layers. Detailed numerical computations are performed to assess the performance and precision of the method.Comment: 13 pages. Typos corrected. Accepted for publication in Physical Review C. A computer code is available on Zenodo at http://doi.org/10.5281/zenodo.3839787 arXiv admin note: text overlap with arXiv:2003.0098

The symmetric eigenvalue problem : stochastic perturbation theory and some network applications

This thesis is concerned with stochastic perturbation theory of the symmetric eigen-value problem. In particular, we provide results about the probability of interchanges in the ordering of the eigenvalues and changes in the eigenvectors of symmetric matrices subject to stochastic perturbations. In this analysis we use a novel combination of traditional Numerical Linear Algebra, Perturbation Theory and Probability Theory. The motivation for this study arises from reliability of spectral clustering of networks, when network data is subject to noise. As far as we are aware, there is nothing comparable in the literature. Further, we make conjectures from which we derive an asymptotic relation between the distributions of the largest eigenvalue and the 2-norm of random symmetric ma- trices, whose entries above the main diagonal are independent, identically distributed random variables with probability density functions being symmetric with respect to zero, including matrices from the Gaussian Orthogonal Ensemble (GOE). As far as we know, some of these conjectures are not new (possibly only as conjectures) but we are not aware of any proofs. Also, we consider networks of coupled oscillators. In their analysis we use both, knowledge of dynamical systems and spectral properties of non-negative matrices. As a result, we present an algorithm, which uncovers the \master-slave" structure of the network. With its help, the analysis of the dynamics and the entrainment of the entire network can be reduced to considering only few of the oscillators, those whose dynamics determine the behaviour of the rest. This can be helpful in large networks exhibiting the \master-slave" structure. Finally, we consider similarities of spectral clustering with respect to di®erent matrices which can be associated with a given network. In particular, we compare clustering of products of Path graphs with respect to two di®erent matrices: the Laplacian and the Normalised Laplacian matrices of the graph. We make the comparison by constructing a Homotopy between two eigenvalue problems and, using some Linear Algebra techniques, we show that the two matrices give similar spectral clusterings when applied to products of Path graphs.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

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

DNA meets the SVD

This paper introduces an important area of computational cell biology where complex, publicly available genomic data is being examined by linear algebra methods, with the aim of revealing biological and medical insights

Primary evolving networks and the comparative analysis of robust and fragile structures

In this paper we consider the structure of dynamically evolving networks modelling information and activity moving across a large set of vertices. We adopt the communicability concept that generalizes that of centrality which is defined for static networks. We define the primary network structure within the whole as comprising of the most influential vertices (both as senders and receivers of dynamically sequenced activity). We present a methodology based on successive vertex knockouts, up to a very small fraction of the whole primary network,that can characterize the nature of the primary network as being either relatively robust and lattice-like (with redundancies built in) or relatively fragile and tree-like (with sensitivities and few redundancies). We apply these ideas to the analysis of evolving networks derived from fMRI scans of resting human brains. We show that the estimation of performance parameters via the structure tests of the corresponding primary networks is subject to less variability than that observed across a very large population of such scans. Hence the differences within the population are significant

Towards the computer-aided diagnosis of dementia based on the geometric and network connectivity of structural MRI data

We present an intuitive geometric approach for analysing the structure and fragility of T1-weighted structural MRI scans of human brains. Apart from computing characteristics like the surface area and volume of regions of the brain that consist of highly active voxels, we also employ Network Theory in order to test how close these regions are to breaking apart. This analysis is used in an attempt to automatically classify subjects into three categories: Alzheimer’s disease, mild cognitive impairment and healthy controls, for the CADDementia Challenge

The effect of high energy milling and high-thermal treatment on the structure and thermal decomposition of minerals from natural CaO-SiO2-P2O5 ceramic system

The major issue studied in this paper is a natural mineral aggregate of quartz, calcite, and fluorapatite (the raw material originated from Bulgaria) before and after high energy milling and thermal treatment, in the order to investigate the properties of natural CaO-SiO2-P2O5 ceramic system. The activation effects are monitored by chemical analysis, X-ray powder diffraction, Fourier transformed infrared spectroscopy, and Thermal analysis (TG/DTG). The activation effect study shows: (i) change of chemical bond strength; (ii) deformation of structural polyhedrons with the formation of new isomorphic phases; (iii) the prolonged time of HEM activation leads to lower raw mineral stability and to the formation of new phases; iv) increased SiO2 reactivity resulting in solid-phase crystallization. The obtained results can be used in the study of ceramic and cement materials (ancient and modern), soil conditioners, etc