Structural and dynamical interdependencies in complex networks at meso- and macroscale: nestedness, modularity, and in-block nestedness

Many real systems like the brain are considered to be complex, i.e. they are made of several interacting components and display a collective behaviour that cannot be inferred from how the individual parts behave. They are usually described as networks, with the components represented as nodes and the interactions between them as links. Research into networks mainly focuses on exploring how a network's dynamic behaviour is constrained by the nature and topology of the interactions between its elements. Analyses of this sort are performed on three scales: the microscale, based on single nodes; the macroscale, which explores the whole network; and the mesoscale, which studies groups of nodes. Nonetheless, most studies so far have focused on only one scale, despite increasing evidence suggesting that networks exhibit structure on several scales. In our thesis, we apply structural analysis to a variety of synthetic and empirical networks on multiple scales. We focus on the examination of nested, modular, and in-block nested patterns, and the effects that they impose on each other. Finally, we introduce a theoretical model to help us to better understand some of the mechanisms that enable such patterns to emerge.Molts sistemes, com el cervell o internet, són considerats complexos: sistemes formats per una gran quantitat d'elements que interactuen entre si, que exhibeixen un comportament col·lectiu que no es pot inferir des de les propietats dels seus elements aïllats. Aquests sistemes s'estudien mitjançant xarxes, en les quals els elements constituents són els nodes, i les interaccions entre ells, els enllaços. La recerca en xarxes s'enfoca principalment a explorar com el comportament dinàmic d'una xarxa està definit per la naturalesa i la topologia de les interaccions entre els seus elements. Aquesta anàlisi sovint es fa en tres escales: la microescala, que estudia les propietats dels nodes individuals; la macroescala, que explora les propietats de tota la xarxa, i la mesoescala, basada en les propietats de grups de nodes. No obstant, la majoria dels estudis se centren només en una escala, tot i la creixent evidència que suggereix que les xarxes sovint exhibeixen estructura a múltiples escales. En aquesta tesi estudiarem les propietats estructurals de les xarxes a escala múltiple. Analitzarem les propietats estructurals dels patrons in-block nested i la seva relació amb els patrons niats i modulars. Finalment, introduirem un model teòric per explorar alguns dels mecanismes que permeten l'emergència d'aquests patrons.Muchos sistemas, como el cerebro o internet, son considerados complejos: sistemas formados por una gran cantidad de elementos que interactúan entre sí, que exhiben un comportamiento colectivo que no puede inferirse desde las propiedades de sus elementos aislados. Estos sistemas se estudian mediante redes, en las que los elementos constituyentes son los nodos, y las interacciones entre ellos, los enlaces. La investigación en redes se enfoca principalmente a explorar cómo el comportamiento dinámico de una red está definido por la naturaleza y la topología de las interacciones entre sus elementos. Este análisis a menudo se hace en tres escalas: la microescala, que estudia las propiedades de los nodos individuales; la macroescala, que explora las propiedades de toda la red, y la mesoescala, basada en las propiedades de grupos de nodos. No obstante, la mayoría de los estudios se centran solo en una escala, a pesar de la creciente evidencia que sugiere que las redes a menudo exhiben estructura a múltiples escalas. En esta tesis estudiaremos las propiedades estructurales de las redes a escala múltiple. Analizaremos las propiedades estructurales de los patrones in-block nested y su relación con los patrones anidados y modulares. Finalmente, introduciremos un modelo teórico para explorar algunos de los mecanismos que permiten la emergencia de estos patrones.Tecnologías de la información y de rede

Perfectly Secure Communication, based on Graph-Topological Addressing in Unique-Neighborhood Networks

We consider network graphs $G=(V,E)$ in which adjacent nodes share common secrets. In this setting, certain techniques for perfect end-to-end security (in the sense of confidentiality, authenticity (implying integrity) and availability, i.e., CIA+) can be made applicable without end-to-end shared secrets and without computational intractability assumptions. To this end, we introduce and study the concept of a unique-neighborhood network, in which nodes are uniquely identifiable upon their graph-topological neighborhood. While the concept is motivated by authentication, it may enjoy wider applicability as being a technology-agnostic (yet topology aware) form of addressing nodes in a network

The structure and function of complex networks

Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.Comment: Review article, 58 pages, 16 figures, 3 tables, 429 references, published in SIAM Review (2003

Phylogenetic Flexibility via Hall-Type Inequalities and Submodularity

Given a collection τ of subsets of a finite set X, we say that τ is phylogenetically flexible if, for any collection R of rooted phylogenetic trees whose leaf sets comprise the collection τ , R is compatible (i.e. there is a rooted phylogenetic X-tree that displays each tree in R). We show that τ is phylogenetically flexible if and only if it satisfies a Hall-type inequality condition of being ‘slim’. Using submodularity arguments, we show that there is a polynomial-time algorithm for determining whether or not τ is slim. This ‘slim’ condition reduces to a simpler inequality in the case where all of the sets in τ have size 3, a property we call ‘thin’. Thin sets were recently shown to be equivalent to the existence of an (unrooted) tree for which the median function provides an injective mapping to its vertex set; we show here that the unrooted tree in this representation can always be chosen to be a caterpillar tree. We also characterise when a collection τ of subsets of size 2 is thin (in terms of the flexibility of total orders rather than phylogenies) and show that this holds if and only if an associated bipartite graph is a forest. The significance of our results for phylogenetics is in providing precise and efficiently verifiable conditions under which supertree methods that require consistent inputs of trees can be applied to any input trees on given subsets of species

4Ward: a Relayering Strategy for Efficient Training of Arbitrarily Complex Directed Acyclic Graphs

Thanks to their ease of implementation, multilayer perceptrons (MLPs) have become ubiquitous in deep learning applications. The graph underlying an MLP is indeed multipartite, i.e. each layer of neurons only connects to neurons belonging to the adjacent layer. In contrast, in vivo brain connectomes at the level of individual synapses suggest that biological neuronal networks are characterized by scale-free degree distributions or exponentially truncated power law strength distributions, hinting at potentially novel avenues for the exploitation of evolution-derived neuronal networks. In this paper, we present ``4Ward'', a method and Python library capable of generating flexible and efficient neural networks (NNs) from arbitrarily complex directed acyclic graphs. 4Ward is inspired by layering algorithms drawn from the graph drawing discipline to implement efficient forward passes, and provides significant time gains in computational experiments with various Erd\H{o}s-R\'enyi graphs. 4Ward not only overcomes the sequential nature of the learning matrix method, by parallelizing the computation of activations, but also addresses the scalability issues encountered in the current state-of-the-art and provides the designer with freedom to customize weight initialization and activation functions. Our algorithm can be of aid for any investigator seeking to exploit complex topologies in a NN design framework at the microscale

Reconstructing networks

Complex networks datasets often come with the problem of missing information: interactions data that have not been measured or discovered, may be affected by errors, or are simply hidden because of privacy issues. This Element provides an overview of the ideas, methods and techniques to deal with this problem and that together define the field of network reconstruction. Given the extent of the subject, the authors focus on the inference methods rooted in statistical physics and information theory. The discussion is organized according to the different scales of the reconstruction task, that is, whether the goal is to reconstruct the macroscopic structure of the network, to infer its mesoscale properties, or to predict the individual microscopic connections