Graphs and networks theory

This chapter discusses graphs and networks theory

A statistical mechanics approach to autopoietic immune networks

The aim of this work is to try to bridge over theoretical immunology and disordered statistical mechanics. Our long term hope is to contribute to the development of a quantitative theoretical immunology from which practical applications may stem. In order to make theoretical immunology appealing to the statistical physicist audience we are going to work out a research article which, from one side, may hopefully act as a benchmark for future improvements and developments, from the other side, it is written in a very pedagogical way both from a theoretical physics viewpoint as well as from the theoretical immunology one. Furthermore, we have chosen to test our model describing a wide range of features of the adaptive immune response in only a paper: this has been necessary in order to emphasize the benefit available when using disordered statistical mechanics as a tool for the investigation. However, as a consequence, each section is not at all exhaustive and would deserve deep investigation: for the sake of completeness, we restricted details in the analysis of each feature with the aim of introducing a self-consistent model.Comment: 22 pages, 14 figur

The RSB order parameter in finite-dimensional spin glasses: numerical computation at zero temperature

This thesis is focused on the computation of the overlap distribution which characterizes spin glasses with finite connectivity upon an RSB transition at zero temperature. Two models are considered: the J± Bethe lattice spin glass and the Edwards-Anderson spin glass in three dimensions with random regular bond dilution (random dilution with the constraint of fixed connectivity z = 3). The approach is based on the study of the effects of a bulk perturbation on the energy landscape. In ultrametric spin glasses, the distribution of the excited states is known to be related to the order parameter through a universal formula. This formula is used for deriving the order parameter from the experimental distributions. In addition, the finite-size corrections to the ground state energy are computed for the two models

Reading the news through its structure: new hybrid connectivity based approaches

In this thesis a solution for the problem of identifying the structure of news published by online newspapers is presented. This problem requires new approaches and algorithms that are capable of dealing with the massive number of online publications in existence (and that will grow in the future). The fact that news documents present a high degree of interconnection makes this an interesting and hard problem to solve. The identification of the structure of the news is accomplished both by descriptive methods that expose the dimensionality of the relations between different news, and by clustering the news into topic groups. To achieve this analysis this integrated whole was studied using different perspectives and approaches. In the identification of news clusters and structure, and after a preparatory data collection phase, where several online newspapers from different parts of the globe were collected, two newspapers were chosen in particular: the Portuguese daily newspaper Público and the British newspaper The Guardian. In the first case, it was shown how information theory (namely variation of information) combined with adaptive networks was able to identify topic clusters in the news published by the Portuguese online newspaper Público. In the second case, the structure of news published by the British newspaper The Guardian is revealed through the construction of time series of news clustered by a kmeans process. After this approach an unsupervised algorithm, that filters out irrelevant news published online by taking into consideration the connectivity of the news labels entered by the journalists, was developed. This novel hybrid technique is based on Qanalysis for the construction of the filtered network followed by a clustering technique to identify the topical clusters. Presently this work uses a modularity optimisation clustering technique but this step is general enough that other hybrid approaches can be used without losing generality. A novel second order swarm intelligence algorithm based on Ant Colony Systems was developed for the travelling salesman problem that is consistently better than the traditional benchmarks. This algorithm is used to construct Hamiltonian paths over the news published using the eccentricity of the different documents as a measure of distance. This approach allows for an easy navigation between published stories that is dependent on the connectivity of the underlying structure. The results presented in this work show the importance of taking topic detection in large corpora as a multitude of relations and connectivities that are not in a static state. They also influence the way of looking at multi-dimensional ensembles, by showing that the inclusion of the high dimension connectivities gives better results to solving a particular problem as was the case in the clustering problem of the news published online.Neste trabalho resolvemos o problema da identificação da estrutura das notícias publicadas em linha por jornais e agências noticiosas. Este problema requer novas abordagens e algoritmos que sejam capazes de lidar com o número crescente de publicações em linha (e que se espera continuam a crescer no futuro). Este facto, juntamente com o elevado grau de interconexão que as notícias apresentam tornam este problema num problema interessante e de difícil resolução. A identificação da estrutura do sistema de notícias foi conseguido quer através da utilização de métodos descritivos que expõem a dimensão das relações existentes entre as diferentes notícias, quer através de algoritmos de agrupamento das mesmas em tópicos. Para atingir este objetivo foi necessário proceder a ao estudo deste sistema complexo sob diferentes perspectivas e abordagens. Após uma fase preparatória do corpo de dados, onde foram recolhidos diversos jornais publicados online optou-se por dois jornais em particular: O Público e o The Guardian. A escolha de jornais em línguas diferentes deve-se à vontade de encontrar estratégias de análise que sejam independentes do conhecimento prévio que se tem sobre estes sistemas. Numa primeira análise é empregada uma abordagem baseada em redes adaptativas e teoria de informação (nomeadamente variação de informação) para identificar tópicos noticiosos que são publicados no jornal português Público. Numa segunda abordagem analisamos a estrutura das notícias publicadas pelo jornal Britânico The Guardian através da construção de séries temporais de notícias. Estas foram seguidamente agrupadas através de um processo de k-means. Para além disso desenvolveuse um algoritmo que permite filtrar de forma não supervisionada notícias irrelevantes que apresentam baixa conectividade às restantes notícias através da utilização de Q-analysis seguida de um processo de clustering. Presentemente este método utiliza otimização de modularidade, mas a técnica é suficientemente geral para que outras abordagens híbridas possam ser utilizadas sem perda de generalidade do método. Desenvolveu-se ainda um novo algoritmo baseado em sistemas de colónias de formigas para solução do problema do caixeiro viajante que consistentemente apresenta resultados melhores que os tradicionais bancos de testes. Este algoritmo foi aplicado na construção de caminhos Hamiltonianos das notícias publicadas utilizando a excentricidade obtida a partir da conectividade do sistema estudado como medida da distância entre notícias. Esta abordagem permitiu construir um sistema de navegação entre as notícias publicadas que é dependente da conectividade observada na estrutura de notícias encontrada. Os resultados apresentados neste trabalho mostram a importância de analisar sistemas complexos na sua multitude de relações e conectividades que não são estáticas e que influenciam a forma como tradicionalmente se olha para sistema multi-dimensionais. Mostra-se que a inclusão desta dimensões extra produzem melhores resultados na resolução do problema de identificar a estrutura subjacente a este problema da publicação de notícias em linha

The Maximum Binary Tree Problem

We introduce and investigate the approximability of the maximum binary tree problem (MBT) in directed and undirected graphs. The goal in MBT is to find a maximum-sized binary tree in a given graph. MBT is a natural variant of the well-studied longest path problem, since both can be viewed as finding a maximum-sized tree of bounded degree in a given graph. The connection to longest path motivates the study of MBT in directed acyclic graphs (DAGs), since the longest path problem is solvable efficiently in DAGs. In contrast, we show that MBT in DAGs is in fact hard: it has no efficient exp(-O(log n/ log log n))-approximation algorithm under the exponential time hypothesis, where n is the number of vertices in the input graph. In undirected graphs, we show that MBT has no efficient exp(-O(log^0.63 n))-approximation under the exponential time hypothesis. Our inapproximability results rely on self-improving reductions and structural properties of binary trees. We also show constant-factor inapproximability assuming P ? NP. In addition to inapproximability results, we present algorithmic results along two different flavors: (1) We design a randomized algorithm to verify if a given directed graph on n vertices contains a binary tree of size k in 2^k poly(n) time. (2) Motivated by the longest heapable subsequence problem, introduced by Byers, Heeringa, Mitzenmacher, and Zervas, ANALCO 2011, which is equivalent to MBT in permutation DAGs, we design efficient algorithms for MBT in bipartite permutation graphs

Co-evolutionary dynamics of networks and play

Diese Dissertation präsentiert drei inhaltlich zusammenhängende Artikel zu Ko-evolutionären Dynamiken von Netzwerken und Spielen. Ein Ko-evolutionärer Prozess besteht aus drei elementaren Ereignissen- Revision von gewählten Aktionen, Kreation eines links, Zerstörung eines links- welche zusammengefasst eine aggregierte Spieldynamik generieren. Unser Augenmerk liegt in der Charakterisierung des asymptotischen Verhaltens derartiger Prozesse. Kapitel 2, "On a general class of stochastic co-evolutionary dynamics", beschreibt den abstrakten mathematischen Rahmen eines Ko-evolutorischen Modells. Die Spieler sind charakterisiert durch eine Nutzenfunktionen auf einem gemeinsamen Aktionenraum und verwenden probabilistische Verhaltensregeln in den eingangs genannten Ereignissen. Zulässige Verhaltensregeln erfüllen eine Irreduzibilitätsannahme sowie ein Prinzip der großen Abweichungen. Neben diesen technischen Annahmen werden substantielle Annahmen an das Verhalten der Spieler auf ein Minimum reduziert. Dies generiert eine wohl definierte Markov-Kette, dessen langfristiges Verhalten durch Graphen-theoretische Methoden nach Freidlin und Wentzell [Random perturbations of dynamical systems, Springer, 1998] studiert werden kann. Wir beschreiben eine allgemeine Methode mit der stochastisch-stabile Zustände identifiziert werden können. Unter weiteren schwachen Annahmen lässt sich das induzierte Zufallsgraphemodell vollständig charakterisieren. Es zeigt sich eine äußerst interessante und unerwartete Beziehung zwischen Ko-evolutorischen Modellen und dem Modell der inhomogenen Zufallsgraphen. Kapitel 4 präsentiert den Artikel "Potential games played in volatile environments". Dort diskutieren wir ein Ko-evolutorisches Modell in der Klasse von Potentialspielen und Logit-Verhaltensregeln. Die invariante Verteilung sowie das generierte Zufallsgraphenmodell sind vollständig bestimmbar. Weiteres werden einige Statistiken des Zufallsgraphemodells in geschlossener Form präsentiert, wie etwa die ``degree distribution'' des Zufallsgraphen. Kapitel 5 erweitert dieses Modell durch Heterogenität in den Präferenzen der Spieler. Wir definieren eine neue Klasse von Spielen, ``structured Bayesian interaction games'', aufbauend auf einer jungen Literatur der evolutionären Spieltheorie die sich mit ``Bayesian population games'' beschäftigt. Kapitel 3 stellt eine Verbindung zwischen den Markov-Ketten von Kapitel 2, und den Markov-Prozessen der Kapitel 4 und 5 her. Ein abschließendes Kapitel resümiert die Ergebnisse der Dissertation und gibt Ausblicke für zukünftige Forschungsvorhaben.This dissertation presents three interrelated papers on the co-evolution of networks and play. The general structure of these models combines three elementary events- action adjustment, link creation and link destruction- to one stochastic game dynamics, and focuses on the asymptotic properties of these processes. Chapter 2 presents the general mathematical framework of a co-evolutionary model. The players have arbitrary utility functions, defined on a common set of actions, and employ probabilistic behavioral rules in the above mentioned events. Admissible rules satisfy irreducibility and a large deviations assumption. Beside these technical assumptions, we try to avoid making substantial behavioral assumptions. This generates a well-defined Markov chain, whose long-run properties can be studied analytically by making use of tree-characterization methods due to Freidlin and Wentzell [Random perturbations of dynamical systems, Springer, 1998]. We provide a general technique to compute stochastically stable states in such co-evolutionary models, by defining suitable cost-functions. Making some mild additional assumptions on the structure of the behavioral rules, we demonstrate an interesting and unforeseen connection between the derived ensemble of networks and inhomogeneous random graphs. The models presented in chapters 4 and 5 particularize the general framework to the class of potential games and behavioral rules of the logit-response form. Under these assumptions, chapter 4 gives a full description of the induced ensemble of networks, and provides additionally closed-form expression for statistics of this ensemble, such as the degree-distribution. The model presented in chapter 5 is more general by allowing the players to have idiosyncratic preferences. This leads us to the definition of structured Bayesian interaction games, following a recent literature of evolutionary game theory studying Bayesian population games. Chapter 3 establishes a connection between the Markov chain constructed in the general framework of chapter 2 and the continuous-time Markov processes considered in the models of chapters 4 and 5. Chapter 6 recapitulates the results of the thesis and gives an outlook for future research projects