Information Dissemination in Random Networks

Die vorliegende Dissertation beschÃ$ftigt sich mit der Dissemination von Information in einem Kommunikationsnetzwerk mit Broadcast-Kanal. Die zentrale Frage, welcher wir uns in dieser Arbeit widmen, ist wie man eine Nachricht ausgehend von einem Quellknoten effizient an alle anderen Knoten im Netzwerk verteilt. Hierbei verfolgen wir zwei Hauptziele: (1) Die Nachricht soll mit hoher Wahrscheinlichkeit alle Knoten im Netzwerk erreichen; (2) Es sollen so wenige Ébertragungen wie mÃglich stattfinden. In diesem Zusammenhang wenden wir uns hauptsÃ$chlich Algorithmen zur probabilistischen Dissemination von Information zu. Wir modellieren Kommunikationsnetzwerke als Zufallsgraphen, die auf stochastischen Prozessen beruhen. Wir verwenden Methoden der Graphentheorie sowie der stochastischen Geometrie um Disseminationsalgorithmen basierend sowohl auf Nachrichtenweiterleitung als auch auf Network Coding zu analysieren. Unser erstes Resultat ist eine analytische Studie von probabilistischem Flooding. In dieser Studie zeigen wir, wie die netzwerkweite Weiterleitungswahrscheinlichkeit gewÃ$hlt werden soll, sodass eine Nachricht mit hoher Wahrscheinlichkeit alle Knoten im Netzwerk erreicht. Als nÃ$chstes widmen wir uns der Frage, welche Vorteile ein probabilistischer Flooding-Algorithmus basierend auf Network Coding gegenÃber klassischen Methoden hat. Dabei wird die Network-Coding Methode mit dem weit verbreiteten MultiPoint Relay-Algorithmus verglichen. Der Vergleich erfolgt mittels analytischer und numerischer Methoden. Schlussendlich verwenden wir die Erkenntnisse der oben beschriebenen Studien dazu, um ein vernetztes Sensor-Aktuator-System zu entwerfen, welches als Notfallschutzsystem innerhalb von GebÃ$uden zum Einsatz kommen soll. Es soll Personen den kuerzesten sicheren Pfad zu den NotausgÃ$ngen anzeigen. Das Auffinden dieser Pfade erfolgt dabei verteilt basierend auf den Messungen der einzelnen Knoten, die Ãber das gesamte Netzwerk disseminiert werden.This dissertation focuses on the study of information dissemination in communication networks with a broadcast medium. The main problem we address is how to disseminate efficiently a message from a source node to all other network nodes. In terms of efficiency we target two goals: (1) to deliver a source message to all network nodes with high probability; and (2) to use as few transmissions as possible for a given target reachability. In this context, our main focus is devoted to probabilistic dissemination algorithms. Modeling networks as random graphs, which are built from stochastic processes, and using methods from graph theory and stochastic geometry we address both replication based and network coded information dissemination approaches. The first contribution is an analytical study of probabilistic flooding which answers the question of which is the minimum common network-wide forwarding probability each node should use such that a flooded message is obtained by all nodes with high probability. Next, we address the question of which benefits can be expected from network coded based probabilistic flooding. We compare these benefits with the ones from the well established replication based MultiPoint Relay flooding. The study of their efficiency is performed both by analytical techniques and numerical methods. Finally, we apply the insights gained from the study of information dissemination algorithms to the design of a sensor-actuator networked system for emergency response in indoor scenarios. The system guides people to the exits of a building via the shortest safe paths, computed autonomously by each node whenever a new measurement collected by a sensor is flooded throughout the network.Sérgio Armindo Lopes CrisóstomoAbweichender Titel laut Übersetzung der Verfasserin/des VerfassersZsfassung in dt. und span. SpracheKlagenfurt, Alpen-Adria-Univ., Diss., 2012OeBB(VLID)241066

A mathematical foundation for the use of cliques in the exploration of data with navigation graphs

Navigation graphs were introduced by Hurley and Oldford (2011) as a graph-theoretic framework for exploring data sets, particularly those with many variables. They allow the user to visualize one small subset of the variables and then proceed to another subset, which shares a few of the original variables, via a smooth transition. These graphs serve as both a high level overview of the dataset as well as a tool for a first-hand exploration of regions deemed interesting. This work examines the nature of cliques in navigation graphs, both in terms of type and magnitude, and speculates as to what their significance to the underlying dataset might be. The questions answered by this body of work were motivated by the belief that the presence of cliques in navigation graphs is a potential indicator for the existence of an interesting, possibly unanticipated, relationship among some of the variables. In this thesis we provide a detailed examination of cliques in navigation graphs, both in terms of type, size and number. The study of types of cliques informs us of the potential significance of highly connected structures to the underlying data and guides our approach for examining the possible clique sizes and counts. On the other hand, the prevalence of large clique sizes and counts is suggestive of an interesting, possibly unexpected, relationship between the variates in the data. To address the challenges surrounding the nature of cliques in navigation graphs, we develop a framework for the derivation of closed-form expressions for the moments of count random variables in terms of their underlying indecomposable summands is established. We use this framework in conjunction with a connection between intersecting set families to obtain edge counts within a clique cover and thus, obtain closed-form expressions for the moments of clique counts in random graphs

Un processus empirique à valeurs mesures pour un système de particules en interaction appliqué aux réseaux complexes

Tableau d’honneur de la Faculté des études supérieures et postdoctorales, 2018-2019On propose dans cette thèse une modélisation des réseaux sociaux par des processus aléatoires à valeurs mesures. Notre démarche se base sur une approche par espace latent. Cette dernière a été utilisée dans la littérature dans le but de décrire des interactions non-observées ou latentes dans la dynamique des réseaux complexes. On caractérise les individus du réseau par des mesures de Dirac représentant leurs positions dans l’espace latent. On obtient ainsi une caractérisation du réseau en temps continu par un processus de Markov à valeurs mesures écrit comme la somme des mesures de Dirac représentant les individus. On associe au réseau trois événements aléatoires simples décrivant les arrivées et les départs d’individus suivant des horloges exponentielles en associant chaque événement à une mesure aléatoire de Poisson. Cette thèse est composée essentiellement d’un premier chapitre réservé à l’état de l’art de la littérature de la modélisation des réseaux complexes suivi d’un second chapitre introductif aux processus aléatoires à valeurs mesures. Le 3-ème et 4-ème chapitres sont constitués de deux articles co-écrits avec mon directeur de thèse, Khader Khadraoui, et sont soumis pour publication dans des journaux. Le premier article, inclus dans le chapitre 3, se compose essentiellement de la description détaillée du modèle proposé ainsi que d’une procédure de Monte Carlo permettant de générer aléatoirement des réalisations du modèle, suivi d’une analyse des propriétés théoriques du processus aléatoire à valeurs mesures sous-jacent. On explicitera notamment le générateur infinitésimal du processus de Markov qui caractérise le réseau. On s’intéressera également aux propriétés de survie et d’extinction du réseau puis on proposera une analyse asymptotique dans laquelle on démontrera, en utilisant des techniques de renormalisation, la convergence faible du processus vers une mesure déterministe solution d’un système intégro-différentiel. On terminera l’article par une étude numérique démontrant par des simulations les principales propriétés obtenues avec notre modèle. Dans le second article, inclus dans le chapitre 4, on reformule notre modèle du point de vue des graphes géométriques aléatoires. Une introduction aux graphes géométriques aléatoires est d’ailleurs proposée au chapitre 1 de cette thèse. Le but de notre démarche est d’étudier les propriétés de connectivité du réseau. Ces problématiques sont largement étudiées dans la littérature des graphes géométriques aléatoires et représentent un intérêt théorique et pratique considérable. L’idée proposée est de considérer notre modèle comme un graphe géométrique aléatoire où l’espace latent représente l’espace sous-jacent et la distribution sous-jacente est celle donnée par le processus génératif du réseau. À partir de là, la question de la connectivité du graphe se pose naturellement. En particulier, on s’intéressera à la distribution des sommets isolés, i.e. d’avoir des membres sans connexion dans le réseau. Pour cela, on pose l’hypothèse supplémentaire que chaque individu dans le graphe peut être actif ou non actif suivant une loi de Bernoulli. On démontrera alors que pour certaines valeurs du seuil de connectivité, le nombre d’individus isolés suit asymptotiquement une loi de Poisson. Aussi, la question de la détection de communautés (clustering) dans leréseau est traitée en fonction du seuil de connectivité établi. Nous terminons cette thèse par une conclusion dans laquelle on discute de la pertinence des approches proposées ainsi que des perspectives que peut offrir notre démarche. En particulier, on donne des éléments permettant de généraliser notre démarche à une classe plus large de réseaux complexes.La fin du document est consacrée aux références bibliographiques utilisées tout au long de ce travail ainsi qu’à des annexes dans lesquelles le lecteur pourra trouver des rappels utiles.This thesis concerns the stochastic modelling of complex networks. In particular, weintroduce a new social network model based on a measure-valued stochastic processes. Individuals in the network are characterized by Dirac measures representing their positions in a virtual latent space of affinities. A continuous time network characterizationis obtained by defining an atomic measure-valued Markov process as the sum of some Dirac measures. We endow the network with a basic dynamic describing the random events of arrivals and departures following Poisson point measures. This thesis is essentially consists of a first introductory chapter to the studied problems of complex networks modelling followed by a second chapter where we present an introduction to the theory of measure-valued stochastic processes. The chapters 3 and 4 are essentially composed of two articles co-written with my thesis advisor, Khader Khadraoui and submitted to journals for publication. The first article, included in chapter 3, mainly concerns the detailed description of the proposed model and a Monte Carlo procedure allowing one to generate synthetic networks. Moreover, analysis of the principal theoretical properties of the models is proposed. In particular, the infinitesimal generator of the Markov process which characterizes the network is established. We also study the survival and extinction properties of the network. Therefore, we propose an asymptotic analysis in which we demonstrate, using a renormalization technique, the weak convergence of the network process towards a deterministic measure solution of an integro-differential system. The article is completed by a numerical study. In the second article, included in chapter 4, we reformulate our model from the point of view of random geometric graphs. An introduction to random geometric graphs framework is proposed in chapter 1. The purpose of our approach is to study the connectivity properties of the network. These issues are widely studied in the literature of random geometric graphs and represent a considerable theoretical and practical interest. The proposed idea is to consider the model as a random geometric graph where the latent space represents the underlying space and the underlying distribution is given by the generative process of the network. Therefore, the question of the connectivity of the graph arises naturally. In particular, we focus on the distribution of isolated vertices, i.e. the members with no connections in the network. To this end, we make the additional hypothesis that each individual in the network can be active or not according to a Bernoulli distribution. We then show that for some values of the connectivity threshold, the number of isolated individuals follows a Poisson distribution. In addition, the question of clustering in the network is discussed and illustrated numerically. We conclude this thesis with a conclusion and perspectives chapter in which we discuss the relevance of the proposed approaches as well as the offered perspectives.The end of the thesis is devoted to the bibliographical references used throughout this work as well as appendices in which the reader can find useful reminders

On some problems in extremal hypergraph theory

In this thesis, we study several problems in extremal (hyper)graph theory. We begin by investigating the problem of subgraph containment in the model of randomly perturbed graphs. In particular, we study the perturbed threshold for the appearance of the square of a Hamilton cycle and the problem of finding pairwise vertex-disjoint triangles. We provide a stability version of these results and we discuss their implications on the perturbed thresholds for 2-universality and for a triangle factor. We then turn to the notion of threshold in the context of transversals in hypergraph collections. Here the question is, given a fixed m-edge hypergraph F, how large the minimum degree of each hypergraph Hi needs to be, so that the hypergraph collection (H1, …, Hm) necessarily contains a transversal copy of F. We prove a widely applicable sufficient condition on F such that the following holds. The needed minimum degree is asymptotically the same as the minimum degree required for a copy of F to appear in each Hi. The condition is general enough to obtain transversal variants of various classical Dirac-type results for (powers of) Hamilton cycles. Finally, we initiate the study of a new variant of the Maker-Breaker positional game, which we call the (1:b) multistage Maker-Breaker game. Starting with a given hypergraph, we play several stages of a usual (1:b) Maker-Breaker game where, in each stage, we shrink the board by keeping only the elements that Maker claimed in the previous stage and updating the collection of winning sets accordingly. The game proceeds until no winning sets remain, and the goal of Maker is to prolong the duration of the game for as many stages as possible. We estimate the maximum duration of the (1:b) multistage Maker-Breaker game for several standard graph games played on the edge set of Kn with biases b subpolynomial in n

Beyond Flatland : exploring graphs in many dimensions

Societies, technologies, economies, ecosystems, organisms, . . . Our world is composed of complex networks—systems with many elements that interact in nontrivial ways. Graphs are natural models of these systems, and scientists have made tremendous progress in developing tools for their analysis. However, research has long focused on relatively simple graph representations and problem specifications, often discarding valuable real-world information in the process. In recent years, the limitations of this approach have become increasingly apparent, but we are just starting to comprehend how more intricate data representations and problem formulations might benefit our understanding of relational phenomena. Against this background, our thesis sets out to explore graphs in five dimensions: descriptivity, multiplicity, complexity, expressivity, and responsibility. Leveraging tools from graph theory, information theory, probability theory, geometry, and topology, we develop methods to (1) descriptively compare individual graphs, (2) characterize similarities and differences between groups of multiple graphs, (3) critically assess the complexity of relational data representations and their associated scientific culture, (4) extract expressive features from and for hypergraphs, and (5) responsibly mitigate the risks induced by graph-structured content recommendations. Thus, our thesis is naturally situated at the intersection of graph mining, graph learning, and network analysis.Gesellschaften, Technologien, Volkswirtschaften, Ökosysteme, Organismen, . . . Unsere Welt besteht aus komplexen Netzwerken—Systemen mit vielen Elementen, die auf nichttriviale Weise interagieren. Graphen sind natürliche Modelle dieser Systeme, und die Wissenschaft hat bei der Entwicklung von Methoden zu ihrer Analyse große Fortschritte gemacht. Allerdings hat sich die Forschung lange auf relativ einfache Graphrepräsentationen und Problemspezifikationen beschränkt, oft unter Vernachlässigung wertvoller Informationen aus der realen Welt. In den vergangenen Jahren sind die Grenzen dieser Herangehensweise zunehmend deutlich geworden, aber wir beginnen gerade erst zu erfassen, wie unser Verständnis relationaler Phänomene von intrikateren Datenrepräsentationen und Problemstellungen profitieren kann. Vor diesem Hintergrund erkundet unsere Dissertation Graphen in fünf Dimensionen: Deskriptivität, Multiplizität, Komplexität, Expressivität, und Verantwortung. Mithilfe von Graphentheorie, Informationstheorie, Wahrscheinlichkeitstheorie, Geometrie und Topologie entwickeln wir Methoden, welche (1) einzelne Graphen deskriptiv vergleichen, (2) Gemeinsamkeiten und Unterschiede zwischen Gruppen multipler Graphen charakterisieren, (3) die Komplexität relationaler Datenrepräsentationen und der mit ihnen verbundenen Wissenschaftskultur kritisch beleuchten, (4) expressive Merkmale von und für Hypergraphen extrahieren, und (5) verantwortungsvoll den Risiken begegnen, welche die Graphstruktur von Inhaltsempfehlungen mit sich bringt. Damit liegt unsere Dissertation naturgemäß an der Schnittstelle zwischen Graph Mining, Graph Learning und Netzwerkanalyse

Hamiltonicity problems in random graphs

In this thesis, we present some of the main results proved by the author while fulfilling his PhD. While we present all the relevant results in the introduction of the thesis, we have chosen to focus on two of the main ones. First, we show a very recent development about Hamiltonicity in random subgraphs of the hypercube, where we have resolved a long standing conjecture dating back to the 1980s. Second, we present some original results about correlations between the appearance of edges in random regular hypergraphs, which have many applications in the study of subgraphs of random regular hypergraphs. In particular, these applications include subgraph counts and property testing