Impact of Symmetries in Graph Clustering

Diese Dissertation beschäftigt sich mit der durch die Automorphismusgruppe definierten Symmetrie von Graphen und wie sich diese auf eine Knotenpartition, als Ergebnis von Graphenclustering, auswirkt. Durch eine Analyse von nahezu 1700 Graphen aus verschiedenen Anwendungsbereichen kann gezeigt werden, dass mehr als 70 % dieser Graphen Symmetrien enthalten. Dies bildet einen Gegensatz zum kombinatorischen Beweis, der besagt, dass die Wahrscheinlichkeit eines zufälligen Graphen symmetrisch zu sein bei zunehmender Größe gegen Null geht. Das Ergebnis rechtfertigt damit die Wichtigkeit weiterer Untersuchungen, die auf mögliche Auswirkungen der Symmetrie eingehen. Bei der Analyse werden sowohl sehr kleine Graphen (10 000 000 Knoten/>25 000 000 Kanten) berücksichtigt. Weiterhin wird ein theoretisches Rahmenwerk geschaffen, das zum einen die detaillierte Quantifizierung von Graphensymmetrie erlaubt und zum anderen Stabilität von Knotenpartitionen hinsichtlich dieser Symmetrie formalisiert. Eine Partition der Knotenmenge, die durch die Aufteilung in disjunkte Teilmengen definiert ist, wird dann als stabil angesehen, wenn keine Knoten symmetriebedingt von der einen in die andere Teilmenge abgebildet werden und dadurch die Partition verändert wird. Zudem wird definiert, wie eine mögliche Zerlegbarkeit der Automorphismusgruppe in unabhängige Untergruppen als lokale Symmetrie interpretiert werden kann, die dann nur Auswirkungen auf einen bestimmten Bereich des Graphen hat. Um die Auswirkungen der Symmetrie auf den gesamten Graphen und auf Partitionen zu quantifizieren, wird außerdem eine Entropiedefinition präsentiert, die sich an der Analyse dynamischer Systeme orientiert. Alle Definitionen sind allgemein und können daher für beliebige Graphen angewandt werden. Teilweise ist sogar eine Anwendbarkeit für beliebige Clusteranalysen gegeben, solange deren Ergebnis in einer Partition resultiert und sich eine Symmetrierelation auf den Datenpunkten als Permutationsgruppe angeben lässt. Um nun die tatsächliche Auswirkung von Symmetrie auf Graphenclustering zu untersuchen wird eine zweite Analyse durchgeführt. Diese kommt zum Ergebnis, dass von 629 untersuchten symmetrischen Graphen 72 eine instabile Partition haben. Für die Analyse werden die Definitionen des theoretischen Rahmenwerks verwendet. Es wird außerdem festgestellt, dass die Lokalität der Symmetrie eines Graphen maßgeblich beeinflusst, ob dessen Partition stabil ist oder nicht. Eine hohe Lokalität resultiert meist in einer stabilen Partition und eine stabile Partition impliziert meist eine hohe Lokalität. Bevor die obigen Ergebnisse beschrieben und definiert werden, wird eine umfassende Einführung in die verschiedenen benötigten Grundlagen gegeben. Diese umfasst die formalen Definitionen von Graphen und statistischen Graphmodellen, Partitionen, endlichen Permutationsgruppen, Graphenclustering und Algorithmen dafür, sowie von Entropie. Ein separates Kapitel widmet sich ausführlich der Graphensymmetrie, die durch eine endliche Permutationsgruppe, der Automorphismusgruppe, beschrieben wird. Außerdem werden Algorithmen vorgestellt, die die Symmetrie von Graphen ermitteln können und, teilweise, auch das damit eng verwandte Graphisomorphie Problem lösen. Am Beispiel von Graphenclustering gibt die Dissertation damit Einblicke in mögliche Auswirkungen von Symmetrie in der Datenanalyse, die so in der Literatur bisher wenig bis keine Beachtung fanden

Batch Testing, Adaptive Algorithms, and Heuristic Applications for Stable Marriage Problems

In this dissertation we focus on different variations of the stable matching (marriage) problem, initially posed by Gale and Shapley in 1962. In this problem, preference lists are used to match n men with n women in such a way that no (man, woman) pair exists that would both prefer each other over their current partners. These two would be considered a blocking pair, preventing a matching from being considered stable. In our research, we study three different versions of this problem. First, we consider batch testing of stable marriage solutions. Gusfield and Irving presented an open problem in their 1989 book The Stable Marriage Problem: Structure and Algorithms\u3c\italic\u3e on whether, given a reasonable amount of preprocessing time, stable matching solutions could be verified in less than O(n^2) time. We answer this question affirmatively, showing an algorithm that will verify k different matchings in O((m + kn) log^2 n) time. Second, we show how the concept of an adaptive algorithm can be used to speed up running time in certain cases of the stable marriage problem where the disorder present in preference lists is limited. While a problem with identical lists can be solved in a trivial O(n) running time, we present an O(n+k) time algorithm where the women have identical preference lists, and the men have preference lists that differ in k positions from a set of identical lists. We also show a visualization program for better understanding the effects of changes in preference lists. Finally, we look at preference list based matching as a heuristic for cost based matching problems. In theory, this method can lead to arbitrarily bad solutions, but through empirical testing on different types of random sources of data, we show how to obtain reasonable results in practice using methods for generating preference lists “asymmetrically” that account for long-term ramifications of short-term decisions. We also discuss several ways to measure the stability of a solution and how this might be used for bicriteria optimization approaches based on both cost and stability

ASSESSMENT OF STOCHASTIC APPROXIMATION METHODS AND OF DEGENERACY DIAGNOSTIC TOOLS IN EXPONENTIAL RANDOM GRAPH MODELS

In recent decades there has been an enormous growth of interest in the notion of social network and the methods of Social Network Analysis (SNA). The methodology developed in the field of network analysis has been categorized into descriptive methods and statistical methods. The statistical methods may be organized into two parts; the first group consists of dyadic and triadic methods which represent statistical models for subgraphs and the second group of statistical models for entire graphs and digraphs. In this work we pay attention to the Exponential Random Graph Models (ERGMs), the statistical models which provide a general framework for modeling dependent data where the dependence can be thought of as a neighborhood effect. The present manuscript is based on two main motivations. Firstly, we are interested to examine model diagnostics and check for degeneracy of ERGMs using different methods and functions. Secondly, we aim to evaluate and compare results obtained for networks of various sizes from three different estimation procedures such as Newton-Raphson, Robbins-Monro and Stepping

Dynamics and steady-state properties of adaptive networks

Tese de doutoramento, Física, Universidade de Lisboa, Faculdade de Ciências, 2013Collective phenomena often arise through structured interactions among a system's constituents. In the subclass of adaptive networks, the interaction structure coevolves with the dynamics it supports, yielding a feedback loop that is common in a variety of complex systems. To understand and steer such systems, modeling their asymptotic regimes is an essential prerequisite. In the particular case of a dynamic equilibrium, each node in the adaptive network experiences a perpetual change in connections and state, while a comprehensive set of measures characterizing the node ensemble are stationary. Furthermore, the dynamic equilibria of a wide class of adaptive networks appear to be unique, as their characteristic measures are insensitive to initial conditions in both state and topology. This work focuses on dynamic equilibria in adaptive networks, and while it does so in the context of two paradigmatic coevolutionary processes, obtained results easily generalize to other dynamics. In the rst part, a low-dimensional framework is elaborated on using the adaptive contact process. A tentative description of the phase diagram and the steady state is obtained, and a parameter region identi ed where asymmetric microscopic dynamics yield a symmetry between node subensembles. This symmetry is accounted for by novel recurrence relations, which predict it for a wide range of adaptive networks. Furthermore, stationary nodeensemble distributions are analytically generated by these relations from one free parameter. Secondly, another analytic framework is put forward that detects and describes dynamic equilibria, while assigning to them general properties that must hold for a variety of adaptive networks. Modeling a single node's evolution in state and connections as a random walk, the ergodic properties of the network process are used to extract node-ensemble statistics from the node's long-term behavior. These statistical measures are composed of a variety of stationary distributions that are related to one another through simple transformations. Applying this fully self-su cient framework, the dynamic equilibria of three di erent avors of the adaptive contact process are subsequently described and compared. Lastly, an asymmetric variant of the coevolutionary voter model is motivated and proposed, and as for the adaptive contact process, a low-dimensional description is given. In a parameter region where a dynamic equilibrium lets the in nite system display perpetual dynamics, this description can be further reduced to a one-dimensional random walk. For nite system sizes, this allows to analytically characterize longevity of the dynamic equilibrium, with results being compared to the symmetric variant of the process. A nontrivial parameter combination is identi ed for which, in the low-dimensional description of the process, the asymmetric coevolutionary model emulates symmetric voter dynamics without topological coevolution. This emerging symmetry is partially con rmed for the full system and subsequently elaborated on. Slightly varying the original asymmetric model, an additional asymptotic regime is shown to occur that coexists with all others and complicates system description.A estrutura das interacções entre os constituintes elementares de um sistema está frequentemente na origem de comportamentos colectivos não triviais. Em redes adaptativas, esta estrutura de interacção evolui a par com a dinâaica que nela assenta, traduzindo uma retroacção que de comum encontrar em vários sistemas complexos. Resultados analíticos sobre os estados assimptóticos destes sistemas são uma peça essencial para a sua compreensão e controlo. O equilíbrio dinâmico de um caso particular de estado assimptótico em que cada nodo da rede adaptativa vai sempre mudando o seu estado e as suas ligações a outros nodos, enquanto que um conjunto de medidas que caracterizam estatisticamente o ensemble dos nodos mantêm valores fixos. Alémm disso, uma classe muito geral de redes adaptativas apresenta equilíbrios dinâmicos que parecem ser únicos, no sentido em que aqueles valores estacionários não dependem das condições iniciais, quer em termos do estados dos nodos quer em termos da topologia da rede.Este trabalho incide no estudo do equilíbrio dinâmiico de redes adaptativas no contexto particular de dois modelos paradigmáticos de coevolação, mas os principais resultados podem ser facilmente generalizados a outros processos. Na primeira parte, revisita-se e desenvolve-se uma abordagem da variante adaptativa do processo de contacto baseada num modelo de baixa dimensão. Obtem-se uma descrição aproximada do diagrama de fases do sistema e do equilíbrio dinâmico, e identifica-se nessa fase uma combinação de parâmetros para a qual a dinâmica microscópica, que de assimétrica nos estados dos nodos, da origem a uma simetria entre os dois subconjuntos de nodos. Esta simetria é explicada através da derivação de relações de recorrência para as distribuições de grau, que a preveêm para uma ampla classe de redes adaptativas. Estas relações permitem também gerar analiticamente as distribuições de grau estacionárias de cada subconjunto de nodos a partir de um parâmetro livre.Na segunda parte, desenvolve-se uma outra abordagem analítica que permite detectar e descrever o equilíbrio dinâmico, a partir de propriedades gerais que se têm que verificar em muitas redes adaptativas. Na base desta abordagem está a descrição do processo estocástico associado à evolução do estado e das ligações de cada nó, e as propriedades ergódicas que permitem obter as estatísticas de ensemble na rede a partir do comportamento a longo termo de um nó. Estas medidas estatísticas podem ser calculadas a partir de várias distribuições estacionárias que se relacionam umas com as outras através de transformações simples. Como aplicação desta abordagem completa, os equilíbrios dinâmicos de três diferentes variantes do processo de contacto adaptativo são descritos e comparados. Finalmente, motiva-se e propõe-se uma variante assimétrica do voter model coevolutivo. A fase activa metastável é tentativamente descrita como uma random walk ao longo de uma variedade lenta, à semelhan ca do que foi feito na literatura para o modelo simétrico, e os resultados para os dois casos são comparados.É identicada uma combinação de parâmetros particular para a qual este modelo assim etrico emula o modelo simétrico em rede fixa, o que é mais um exemplo da simetria emergente prevista pelas relações de recorrência estabelecidas na primeira parte. Considera-se ainda uma outra variante assimétrica, mais complexa, do voter model co-evolutivo, que apresenta um diagrama de fases essencialmente diferente, e cuja descrição se mostra requerer novas abordagens.Fundação para a Ciência e a Tecnologia (FCT, SFRH/BD/45179/2008

Dynamics of Patterns

Patterns and nonlinear waves arise in many applications. Mathematical descriptions and analyses draw from a variety of fields such as partial differential equations of various types, differential and difference equations on networks and lattices, multi-particle systems, time-delayed systems, and numerical analysis. This workshop brought together researchers from these diverse areas to bridge existing gaps and to facilitate interaction

Techniques of replica symmetry breaking and the storage problem of the McCulloch-Pitts neuron

In this article the framework for Parisi's spontaneous replica symmetry breaking is reviewed, and subsequently applied to the example of the statistical mechanical description of the storage properties of a McCulloch-Pitts neuron. The technical details are reviewed extensively, with regard to the wide range of systems where the method may be applied. Parisi's partial differential equation and related differential equations are discussed, and a Green function technique introduced for the calculation of replica averages, the key to determining the averages of physical quantities. The ensuing graph rules involve only tree graphs, as appropriate for a mean-field-like model. The lowest order Ward-Takahashi identity is recovered analytically and is shown to lead to the Goldstone modes in continuous replica symmetry breaking phases. The need for a replica symmetry breaking theory in the storage problem of the neuron has arisen due to the thermodynamical instability of formerly given solutions. Variational forms for the neuron's free energy are derived in terms of the order parameter function x(q), for different prior distribution of synapses. Analytically in the high temperature limit and numerically in generic cases various phases are identified, among them one similar to the Parisi phase in the Sherrington-Kirkpatrick model. Extensive quantities like the error per pattern change slightly with respect to the known unstable solutions, but there is a significant difference in the distribution of non-extensive quantities like the synaptic overlaps and the pattern storage stability parameter. A simulation result is also reviewed and compared to the prediction of the theory.Comment: 103 Latex pages (with REVTeX 3.0), including 15 figures (ps, epsi, eepic), accepted for Physics Report

Techniques of replica symmetry breaking and the storage problem of the McCulloch-Pitts neuron

In this article the framework for Parisi's spontaneous replica symmetry breaking is reviewed, and subsequently applied to the example of the statistical mechanical description of the storage properties of a McCulloch-Pitts neuron. The technical details are reviewed extensively, with regard to the wide range of systems where the method may be applied. Parisi's partial differential equation and related differential equations are discussed, and a Green function technique introduced for the calculation of replica averages, the key to determining the averages of physical quantities. The ensuing graph rules involve only tree graphs, as appropriate for a mean-field-like model. The lowest order Ward-Takahashi identity is recovered analytically and is shown to lead to the Goldstone modes in continuous replica symmetry breaking phases. The need for a replica symmetry breaking theory in the storage problem of the neuron has arisen due to the thermodynamical instability of formerly given solutions. Variational forms for the neuron's free energy are derived in terms of the order parameter function x(q), for different prior distribution of synapses. Analytically in the high temperature limit and numerically in generic cases various phases are identified, among them one similar to the Parisi phase in the Sherrington-Kirkpatrick model. Extensive quantities like the error per pattern change slightly with respect to the known unstable solutions, but there is a significant difference in the distribution of non-extensive quantities like the synaptic overlaps and the pattern storage stability parameter. A simulation result is also reviewed and compared to the prediction of the theory.Comment: 103 Latex pages (with REVTeX 3.0), including 15 figures (ps, epsi, eepic), accepted for Physics Report

The Biglobal Instability of the Bidirectional Vortex

State of the art research in hydrodynamic stability analysis has moved from classic one-dimensional methods such as the local nonparallel approach and the parabolized stability equations to two-dimensional, biglobal, methods. The paradigm shift toward two dimensional techniques with the ability to accommodate fully three-dimensional base flows is a necessary step toward modeling complex, multidimensional flowfields in modern propulsive applications. Here, we employ a two-dimensional spatial waveform with sinusoidal temporal dependence to reduce the three-dimensional linearized Navier-Stokes equations to their biglobal form. Addressing hydrodynamic stability in this way circumvents the restrictive parallel-flow assumption and admits boundary conditions in the streamwise direction. Furthermore, the following work employs a full momentum formulation, rather than the reduced streamfunction form, accounting for a nonzero tangential mean flow velocity. This approach adds significant complexity in both formulation and implementation but renders a more general methodology applicable to a broader spectrum of mean flows. Specifically, we consider the stability of three models for bidirectional vortex flow. While a complete parametric study ensues, the stabilizing effect of the swirl velocity is evident as the injection parameter, kappa, is closely examined