Using Pareto optimality to explore the topology and dynamics of the human connectome.

Graph theory has provided a key mathematical framework to analyse the architecture of human brain networks. This architecture embodies an inherently complex relationship between connection topology, the spatial arrangement of network elements, and the resulting network cost and functional performance. An exploration of these interacting factors and driving forces may reveal salient network features that are critically important for shaping and constraining the brain's topological organization and its evolvability. Several studies have pointed to an economic balance between network cost and network efficiency with networks organized in an 'economical' small-world favouring high communication efficiency at a low wiring cost. In this study, we define and explore a network morphospace in order to characterize different aspects of communication efficiency in human brain networks. Using a multi-objective evolutionary approach that approximates a Pareto-optimal set within the morphospace, we investigate the capacity of anatomical brain networks to evolve towards topologies that exhibit optimal information processing features while preserving network cost. This approach allows us to investigate network topologies that emerge under specific selection pressures, thus providing some insight into the selectional forces that may have shaped the network architecture of existing human brains.

Modelització i simulació de processos dinàmics en xarxes complexes adaptatives

El 1736, Leonhard Euler va ser pioner en l'estudi de la teoria de grafs, i des de llavorsmúltiples autors com Kirchoff, Seymour, etc. continuaren amb l'estudi de la teoria i topologiade grafs. La teoria de xarxes, part de la teoria de grafs, també ha estat estudiada abastament.D'altra banda, la dinàmica de xarxes fou popularitzada per Dan Gillespie el 1977, en el qual proposà un algorisme que permet la simulació discreta i estocàstica d'un sistema de partícules, el qual és la base del treball ja que serveix per dur a terme les simulacions de processos sobre les xarxes complexes. El camp de l'anàlisi de la dinàmica de xarxes, de fet, és un campemergent en l'actualitat; comprèn tant l'anàlisi estadística com la utilització de simulacions persolucionar problemes de la mateixa dinàmica.Les xarxes complexes (xarxes de característiques complexes, sovint xarxes reals) també sónobjecte d'estudi de l'actualitat, sobretot a causa de l'aparició de les xarxes socials. S'han convertiten un paradigma per l'estudi de processos dinàmics en sistemes formats per molts componentsque interactuen entre si de manera molt homogèniaL'objectiu del treball és triple:1. Estudiar i entendre els conceptes bàsics i la topologia de les xarxes complexes, així comdiferents tipus de dinàmiques de processos sobre elles.2. Programar un simulador estocàstic en llenguatge C++ capaç de generar trajectòries mitjantçant l'algorisme de Gillespie tant pel model epidèmic com pel model de dinàmicad'enllaços amb reconnexió.3. Utilitzar el simulador tant per estudiar casos que ja han estat tractats en la literatura comcasos nous que no han estat tractats i que poden ser assimilables a xarxes reals com, perexemple, xarxes socials

Sherali-Adams Relaxations and Indistinguishability in Counting Logics

Two graphs with adjacency matrices $\mathbf{A}$ and $\mathbf{B}$ are isomorphic if there exists a permutation matrix $\mathbf{P}$ for which the identity $\mathbf{P}^{\mathrm{T}} \mathbf{A} \mathbf{P} = \mathbf{B}$ holds. Multiplying through by $\mathbf{P}$ and relaxing the permutation matrix to a doubly stochastic matrix leads to the linear programming relaxation known as fractional isomorphism. We show that the levels of the Sherali--Adams (SA) hierarchy of linear programming relaxations applied to fractional isomorphism interleave in power with the levels of a well-known color-refinement heuristic for graph isomorphism called the Weisfeiler--Lehman algorithm, or, equivalently, with the levels of indistinguishability in a logic with counting quantifiers and a bounded number of variables. This tight connection has quite striking consequences. For example, it follows immediately from a deep result of Grohe in the context of logics with counting quantifiers that a fixed number of levels of SA suffice to determine isomorphism of planar and minor-free graphs. We also offer applications in both finite model theory and polyhedral combinatorics. First, we show that certain properties of graphs, such as that of having a flow circulation of a prescribed value, are definable in the infinitary logic with counting with a bounded number of variables. Second, we exploit a lower bound construction due to Cai, Fürer, and Immerman in the context of counting logics to give simple explicit instances that show that the SA relaxations of the vertex-cover and cut polytopes do not reach their integer hulls for up to $\Omega(n)$ levels, where $n$ is the number of vertices in the graph.

Intrahemispheric cortico-cortical connections of the human auditory cortex.

The human auditory cortex comprises the supratemporal plane and large parts of the temporal and parietal convexities. We have investigated the relevant intrahemispheric cortico-cortical connections using in vivo DSI tractography combined with landmark-based registration, automatic cortical parcellation and whole-brain structural connection matrices in 20 right-handed male subjects. On the supratemporal plane, the pattern of connectivity was related to the architectonically defined early-stage auditory areas. It revealed a three-tier architecture characterized by a cascade of connections from the primary auditory cortex to six adjacent non-primary areas and from there to the superior temporal gyrus. Graph theory-driven analysis confirmed the cascade-like connectivity pattern and demonstrated a strong degree of segregation and hierarchy within early-stage auditory areas. Putative higher-order areas on the temporal and parietal convexities had more widely spread local connectivity and long-range connections with the prefrontal cortex; analysis of optimal community structure revealed five distinct modules in each hemisphere. The pattern of temporo-parieto-frontal connectivity was partially asymmetrical. In conclusion, the human early-stage auditory cortical connectivity, as revealed by in vivo DSI tractography, has strong similarities with that of non-human primates. The modular architecture and hemispheric asymmetry in higher-order regions is compatible with segregated processing streams and lateralization of cognitive functions.

Spectral properties of the Laplacian of multiplex networks.

One of the more challenging tasks in the understanding of dynamical properties of models on top of complex networks is to capture the precise role of multiplex topologies. In a recent paper, Gómez et al. [ Phys. Rev. Lett. 110 028701 (2013)], some of the authors proposed a framework for the study of diffusion processes in such networks. Here, we extend the previous framework to deal with general configurations in several layers of networks and analyze the behavior of the spectrum of the Laplacian of the full multiplex. We derive an interesting decoupling of the problem that allow us to unravel the role played by the interconnections of the multiplex in the dynamical processes on top of them. Capitalizing on this decoupling we perform an asymptotic analysis that allow us to derive analytical expressions for the full spectrum of eigenvalues. This spectrum is used to gain insight into physical phenomena on top of multiplex, specifically, diffusion processes and synchronizability.

A structural bridge between alternant and non-alternant hydrocarbons

A simple set of trimethylene-substituted even, fully-pi-bonded, non-alternant monocycles is shown to have several key features in common with acyclic, even alternant polyenes at the Hückel level. These non-alternant molecules provide a bridge between alternant and non-alternant hydrocarbons. This topic might serve as a useful addition to Hückel theory courses targeted at senior undergraduate students.

DWI and complex brain network analysis predicts vascular cognitive impairment in spontaneous hypertensive rats undergoing executive function tests

The identification of biomarkers of vascular cognitive impairment is urgent for its early diagnosis. The aim of this study was to detect and monitor changes in brain structure and connectivity, and to correlate them with the decline in executive function. We examined the feasibility of early diagnostic magnetic resonance imaging (MRI) to predict cognitive impairment before onset in an animal model of chronic hypertension: Spontaneously Hypertensive Rats. Cognitive performance was tested in an operant conditioning paradigm that evaluated learning, memory, and behavioral flexibility skills. Behavioral tests were coupled with longitudinal diffusion weighted imaging acquired with 126 diffusion gradient directions and 0.3 mm(3) isometric resolution at 10, 14, 18, 22, 26, and 40 weeks after birth. Diffusion weighted imaging was analyzed in two different ways, by regional characterization of diffusion tensor imaging (DTI) indices, and by assessing changes in structural brain network organization based on Q-Ball tractography. Already at the first evaluated times, DTI scalar maps revealed significant differences in many regions, suggesting loss of integrity in white and gray matter of spontaneously hypertensive rats when compared to normotensive control rats. In addition, graph theory analysis of the structural brain network demonstrated a significant decrease of hierarchical modularity, global and local efficacy, with predictive value as shown by regional three-fold cross validation study. Moreover, these decreases were significantly correlated with the behavioral performance deficits observed at subsequent time points, suggesting that the diffusion weighted imaging and connectivity studies can unravel neuroimaging alterations even overt signs of cognitive impairment become apparent.

Efficient Optimization Algorithms for Nonlinear Data Analysis

Identification of low-dimensional structures and main sources of variation from multivariate data are fundamental tasks in data analysis. Many methods aimed at these tasks involve solution of an optimization problem. Thus, the objective of this thesis is to develop computationally efficient and theoretically justified methods for solving such problems. Most of the thesis is based on a statistical model, where ridges of the density estimated from the data are considered as relevant features. Finding ridges, that are generalized maxima, necessitates development of advanced optimization methods. An efficient and convergent trust region Newton method for projecting a point onto a ridge of the underlying density is developed for this purpose. The method is utilized in a differential equation-based approach for tracing ridges and computing projection coordinates along them. The density estimation is done nonparametrically by using Gaussian kernels. This allows application of ridge-based methods with only mild assumptions on the underlying structure of the data. The statistical model and the ridge finding methods are adapted to two different applications. The first one is extraction of curvilinear structures from noisy data mixed with background clutter. The second one is a novel nonlinear generalization of principal component analysis (PCA) and its extension to time series data. The methods have a wide range of potential applications, where most of the earlier approaches are inadequate. Examples include identification of faults from seismic data and identification of filaments from cosmological data. Applicability of the nonlinear PCA to climate analysis and reconstruction of periodic patterns from noisy time series data are also demonstrated. Other contributions of the thesis include development of an efficient semidefinite optimization method for embedding graphs into the Euclidean space. The method produces structure-preserving embeddings that maximize interpoint distances. It is primarily developed for dimensionality reduction, but has also potential applications in graph theory and various areas of physics, chemistry and engineering. Asymptotic behaviour of ridges and maxima of Gaussian kernel densities is also investigated when the kernel bandwidth approaches infinity. The results are applied to the nonlinear PCA and to finding significant maxima of such densities, which is a typical problem in visual object tracking.

Linguistic models for decision support

Linguistic modelling is a rather new branch of mathematics that is still undergoing rapid development. It is closely related to fuzzy set theory and fuzzy logic, but knowledge and experience from other fields of mathematics, as well as other fields of science including linguistics and behavioral sciences, is also necessary to build appropriate mathematical models. This topic has received considerable attention as it provides tools for mathematical representation of the most common means of human communication - natural language. Adding a natural language level to mathematical models can provide an interface between the mathematical representation of the modelled system and the user of the model - one that is sufficiently easy to use and understand, but yet conveys all the information necessary to avoid misinterpretations. It is, however, not a trivial task and the link between the linguistic and computational level of such models has to be established and maintained properly during the whole modelling process. In this thesis, we focus on the relationship between the linguistic and the mathematical level of decision support models. We discuss several important issues concerning the mathematical representation of meaning of linguistic expressions, their transformation into the language of mathematics and the retranslation of mathematical outputs back into natural language. In the first part of the thesis, our view of the linguistic modelling for decision support is presented and the main guidelines for building linguistic models for real-life decision support that are the basis of our modeling methodology are outlined. From the theoretical point of view, the issues of representation of meaning of linguistic terms, computations with these representations and the retranslation process back into the linguistic level (linguistic approximation) are studied in this part of the thesis. We focus on the reasonability of operations with the meanings of linguistic terms, the correspondence of the linguistic and mathematical level of the models and on proper presentation of appropriate outputs. We also discuss several issues concerning the ethical aspects of decision support - particularly the loss of meaning due to the transformation of mathematical outputs into natural language and the issue or responsibility for the final decisions. In the second part several case studies of real-life problems are presented. These provide background and necessary context and motivation for the mathematical results and models presented in this part. A linguistic decision support model for disaster management is presented here – formulated as a fuzzy linear programming problem and a heuristic solution to it is proposed. Uncertainty of outputs, expert knowledge concerning disaster response practice and the necessity of obtaining outputs that are easy to interpret (and available in very short time) are reflected in the design of the model. Saaty’s analytic hierarchy process (AHP) is considered in two case studies - first in the context of the evaluation of works of art, where a weak consistency condition is introduced and an adaptation of AHP for large matrices of preference intensities is presented. The second AHP case-study deals with the fuzzified version of AHP and its use for evaluation purposes – particularly the integration of peer-review into the evaluation of R&D outputs is considered. In the context of HR management, we present a fuzzy rule based evaluation model (academic faculty evaluation is considered) constructed to provide outputs that do not require linguistic approximation and are easily transformed into graphical information. This is achieved by designing a specific form of fuzzy inference. Finally the last case study is from the area of humanities - psychological diagnostics is considered and a linguistic fuzzy model for the interpretation of outputs of multidimensional questionnaires is suggested. The issue of the quality of data in mathematical classification models is also studied here. A modification of the receiver operating characteristics (ROC) method is presented to reflect variable quality of data instances in the validation set during classifier performance assessment. Twelve publications on which the author participated are appended as a third part of this thesis. These summarize the mathematical results and provide a closer insight into the issues of the practicalapplications that are considered in the second part of the thesis.

Properties and algorithms of the (n, k)-arrangement graphs

The (n, k)-arrangement interconnection topology was first introduced in 1992. The (n, k )-arrangement graph is a class of generalized star graphs. Compared with the well known n-star, the (n, k )-arrangement graph is more flexible in degree and diameter. However, there are few algorithms designed for the (n, k)-arrangement graph up to present. In this thesis, we will focus on finding graph theoretical properties of the (n, k)- arrangement graph and developing parallel algorithms that run on this network. The topological properties of the arrangement graph are first studied. They include the cyclic properties. We then study the problems of communication: broadcasting and routing. Embedding problems are also studied later on. These are very useful to develop efficient algorithms on this network. We then study the (n, k )-arrangement network from the algorithmic point of view. Specifically, we will investigate both fundamental and application algorithms such as prefix sums computation, sorting, merging and basic geometry computation: finding convex hull on the (n, k )-arrangement graph. A literature review of the state-of-the-art in relation to the (n, k)-arrangement network is also provided, as well as some open problems in this area.

Square Root Finding In Graphs

Abstract: Root and root finding are concepts familiar to most branches of mathematics. In graph theory, H is a square root of G and G is the square of H if two vertices x,y have an edge in G if and only if x,y are of distance at most two in H. Graph square is a basic operation with a number of results about its properties in the literature. We study the characterization and recognition problems of graph powers. There are algorithmic and computational approaches to answer the decision problem of whether a given graph is a certain power of any graph. There are polynomial time algorithms to solve this problem for square of graphs with girth at least six while the NP-completeness is proven for square of graphs with girth at most four. The girth-parameterized problem of root fining has been open in the case of square of graphs with girth five. We settle the conjecture that recognition of square of graphs with girth 5 is NP-complete. This result is providing the complete dichotomy theorem for square root finding problem.

Acyclic 5-Choosability of Planar Graphs Without Adjacent Short Cycles

The conjecture claiming that every planar graph is acyclic 5-choosable[Borodin et al., 2002] has been verified for several restricted classes of planargraphs. Recently, O. V. Borodin and A. O. Ivanova, [Journal of Graph Theory,68(2), October 2011, 169-176], have shown that a planar graph is acyclically 5-choosable if it does not contain an i-cycle adjacent to a j-cycle, where 3<=j<=5 if i=3 and 4<=j<=6 if i=4. We improve the above mentioned result and prove that every planar graph without an i-cycle adjacent to a j-cycle with3<=j<=5 if i=3 and 4<=j<=5 if i=4 is acyclically 5-choosable.

Backbone Colouring of Graphs

Consider an undirected graph G and a subgraph of G, H. A q-backbone k-colouring of (G,H) is a mapping f: V(G) {1, 2, ..., k} such that G is properly coloured and for each edge of H, the colours of its endpoints differ by at least q. The minimum number k for which there is a backbone k-colouring of (G,H) is the backbone chromatic number, BBCq(G,H). It has been proved that backbone k-colouring of (G,T) is at most 4 if G is a connected C4-free planar graph or non-bipartite C5-free planar graph or Cj-free, j∈{6,7,8} planar graph without adjacent triangles. In this thesis we improve the results mentioned above and prove that 2-backbone k-colouring of any connected planar graphs without adjacent triangles is at most 4 by using a discharging method. In the second part of this thesis we further improve these results by proving that for any graph G with χ(G) ≥ 4, BBC(G,T) = χ(G). In fact, we prove the stronger result that a backbone tree T in G exists, such that ∀ uv ∈ T, |f(u)-f(v)|=2 or |f(u)-f(v)| ≥ k-2, k = χ(G). For the case that G is a planar graph, according to Four Colour Theorem, χ(G) = 4; so, BBC(G,T) = 4.

Properties and Algorithms of the KCube Interconnection Networks

The KCube interconnection network was first introduced in 2010 in order to exploit the good characteristics of two well-known interconnection networks, the hypercube and the Kautz graph. KCube links up multiple processors in a communication network with high density for a fixed degree. Since the KCube network is newly proposed, much study is required to demonstrate its potential properties and algorithms that can be designed to solve parallel computation problems. In this thesis we introduce a new methodology to construct the KCube graph. Also, with regard to this new approach, we will prove its Hamiltonicity in the general KC(m; k). Moreover, we will find its connectivity followed by an optimal broadcasting scheme in which a source node containing a message is to communicate it with all other processors. In addition to KCube networks, we have studied a version of the routing problem in the traditional hypercube, investigating this problem: whether there exists a shortest path in a Qn between two nodes 0n and 1n, when the network is experiencing failed components. We first conditionally discuss this problem when there is a constraint on the number of faulty nodes, and subsequently introduce an algorithm to tackle the problem without restrictions on the number of nodes.