[pt] Na primeira parte dessa tese, estudamos o desvio no número de triângulos com respeito à média em ambos os modelos de grafos aleatórios G(n,m) e G(n, p). Focamos no caso em que o grafo aleatório é esparso, no qual a densidade de arestas vai para zero quando o número de vértices cresce para o infinito. Nosso foco também reside no caso de desvios moderados, i.e., aqueles cuja ordem está entre o desvio padrão e a média. Além disso, também derivamos o mesmo tipo de resultado para cerejas (caminhos de comprimento dois). Na segunda parte dessa tese, estudamos a desigualdade de Freedman. Essa desigualdade fornece limitantes para a probabilidade de desvio de um martingal limitado usando sua variância condicional. No nosso trabalho, obtemos uma versão mais forte da desigualdade de Freedman, impondo condições adicionais de simetria nos incrementos do processo martingal. / [en] In the first part of this thesis, we study the deviation of the number of triangles with respect to its mean in both the random graph models G(n,m) and G(n, p). We focus on the case where the random graph is sparse, in which the edge density goes to zero as the number of vertices increases to infinity. Also, our focus is in the case of moderate deviations, i.e., those of order in between the standard deviation and the mean. In addition, we derive the same kind of results for cherries (paths of length two). In the second part of this thesis, we study Freedman s inequality. This inequality gives bounds on the probability of the deviation of a bounded martingale using its conditional variance. In our work, we obtain a strengthening of Freedman s inequality, under additional symmetry conditions on the increments of the martingale process.

No description available.

A central problem in neuroscience is to understand how memories are formed as a result of the activities of neurons. Valiant’s neuroidal model attempted to address this question by modeling the brain as a random graph and memories as subgraphs within that graph. However the question of memory capacity within that model has not been explored: how many memories can the brain hold? Valiant introduced the concept of interference between memories as the defining factor for capacity; excessive interference signals the model has reached capacity. Since then, exploration of capacity has been limited, but recent investigations have delved into the capacity of the Assembly Calculus, a derivative of Valiant's Neuroidal model. In this paper, we provide rigorous definitions for capacity and interference and present theoretical formulations for the memory capacity within a finite set, where subsets represent memories. We propose that these results can be adapted to suit both the Neuroidal model and Assembly calculus. Furthermore, we substantiate our claims by providing simulations that validate the theoretical findings. Our study aims to contribute essential insights into the understanding of memory capacity in complex cognitive models, offering potential ideas for applications and extensions to contemporary models of cognition.

Random graph processes are basic mathematical models for large-scale networks evolving over time. Their systematic study was pioneered by Erdös and Rényi around 1960, and one key feature of many 'classical' models is that the edges appear independently. While this makes them amenable to a rigorous analysis, it is desirable, both mathematically and in terms of applications, to understand more complicated situations. In this thesis the main goal is to improve our rigorous understanding of evolving random graphs with significant dependencies. The first model we consider is known as an Achlioptas process: in each step two random edges are chosen, and using a given rule only one of them is selected and added to the evolving graph. Since 2000 a large class of 'complex' rules has eluded a rigorous analysis, and it was widely believed that these could give rise to a striking and unusual phenomenon. Making this explicit, Achlioptas, D'Souza and Spencer conjectured in Science that one such rule yields a very abrupt (discontinuous) percolation phase transition. We disprove this, showing that the transition is in fact continuous for all Achlioptas process. In addition, we give the first rigorous analysis of the more 'complex' rules, proving that certain key statistics are tightly concentrated (i) in the subcritical evolution, and (ii) also later on if an associated system of differential equations has a unique solution. The second model we study is the H-free process, where random edges are added subject to the constraint that they do not complete a copy of some fixed graph H. The most important open question for such 'constrained' processes is due to Erdös, Suen and Winkler: in 1995 they asked what the typical final number of edges is. While Osthus and Taraz answered this in 2000 up to logarithmic factors for a large class of graphs H, more precise bounds are only known for a few special graphs. We close this gap for the cases where a cycle of fixed length is forbidden, determining the final number of edges up to constants. Our result not only establishes several conjectures, it is also the first which answers the more than 15-year old question of Erdös et. al. for a class of forbidden graphs H.

In this work we study two random procedures in cubic graphs with large girth. The first procedure finds a probability distribution on edge-cuts such that each edge is in a randomly chosen cut with probability at least 0.88672. As corollaries, we derive lower bounds for the size of maximum cut in cubic graphs with large girth and in random cubic graphs, and also an upper bound for the fractional cut covering number in cubic graphs with large girth. The second procedure finds a probability distribution on independent sets such that each vertex is in an independent set with probability at least 0.4352. This implies lower bounds for the size of maximum independent set in cubic graphs with large girth and in random cubic graphs, as well as an upper bound for the fractional chromatic number in cubic graphs with large girth.

Cette thèse est consacrée à l'étude du comportement asymptotique de grands graphes et arbres aléatoires. Le premier modèle étudié est un modèle de graphe aléatoire inhomogène introduit par Bo Söderberg. Un chapitre de ce manuscrit est consacré à l'étude asymptotique de la taille des composantes connexes à proximité de la fenêtre critique, en le reliant à la longueur des excursions d'un mouvement brownien avec dérive parabolique, étendant les résultats obtenus par Aldous. Le chapitre suivant est consacré à un processus de graphes aléatoires proposé par Itai Benjamini, défini ainsi : les arêtes sont ajoutées indépendamment, à taux fixe. Lorsqu'un sommet atteint le degré k, toutes les arêtes adjacentes à ce sommet sont immédiatement supprimées. Ce processus n'est pas croissant, ce qui empêche d'utiliser directement certaines approches usuelles. L'utilisation de limites locales permet de montrer la présence (resp. l'absence) d'une composante géante à certaines étapes dans le cas k>=5 (resp. k<=3). Dans le cas k=4, ces résultats permettent de caractériser la présence d'une composante géante en fonction du caractère surcritique ou non d'un processus de branchement associé. Dans le dernier chapitre est étudiée la hauteur d'un arbre de Lyndon associé à un mot de Lyndon choisi uniformément parmi les mots de Lyndon de longueur n, prouvant que cette hauteur est approximativement c ln n, avec c=5,092... la solution d'un problème d'optimisation. Afin d'obtenir ce résultat, nous couplons d'abord l'arbre de Lyndon à un arbre de Yule, que nous étudions ensuite à l'aide de techniques provenant des théories des marches branchantes et des grandes déviations. / This thesis is dedicated to the study of the asymptotic behavior of some large random graphs and trees. First is studied a random graph model introduced by Bo Söderberg in 2002. One chapter of this manuscript is devoted to the study of the asymptotic behavior of the size of the connected components near the critical window, linking it to the lengths of excursion of a Brownian motion with parabolic drift. The next chapter talks about a random graph process suggested by Itai Benjamini, defined as follows: edges are independently added at a fixe rate. Whenever a vertex reaches degree k, all adjacent edges are removed. This process is non-increasing, preventing the use of some commonly used methods. By using local limits, in the spirit of the PWIT, we were able to prove the presence (resp. absence) of a giant component at some stages of the process when k>=5 (resp. k<=3). In the case k=4, these results allows to link the presence (resp. absence) of a giant component to the supercriticality (resp. criticality or subcriticality) of an associated branching process. In the last chapter, the height of random Lyndon tree is studied, and is proven to be approximately c ln n, in which c=5.092... the solution of an optimization problem. To obtain this result, we couple the Lyndon tree with a Yule tree, then studied with the help of branching walks and large deviations

Diese Habilitationsschrift richtete auf zwei diskrete Strukturen aus: planare Strukturen und zufällige Graphen-Prozesse. Zunächst werden zufällige planare Strukturen untersucht, mit folgende Gesichtspunkte: - Wieviele planare Strukturen gibt es? - Wie kann effizient eine zufällige planare Struktur gleichverteilt erzeugt werden? - Welche asymptotischen Eigenschaften hat eine zufällige planare Struktur mit hoher Wahrscheinlichkeit? Um diese Fragen zu beantworten, werden die planaren Strukturen in Teile mit höherer Konnektivität zerlegt. Für die asymptotische Enumeration wird zuerst die Zerlegung als das Gleichungssystem der generierenden Funktionen interpretiert. Auf dem Gleichungssystem wird dann Singularitätenanalyse angewendet. Für die exakte Enumeration und zufällige Erzeugung wird die rekursive Methode verwendet. Für die typischen Eigenschaften wird die probabilistische Methode auf asymptotischer Anzahl angewendet. Des Weiteren werden zufällige Graphen-Prozesse untersucht. Zufällige Graphen wurden zuerst von Erdos und Renyi eingeführt und untersucht weitgehend seitdem. Ein zufälliger Graphen-Prozess ist eine Markov-Kette, deren Zustandsraum eine Menge der Graphen mit einer gegebenen Knotenmenge ist. Der Prozess fängt mit isolierten Konten an, und in jedem Ablaufschritt entsteht ein neuer Graph aus dem aktuellen Graphen durch das Hinzufügen einer neuen Kante entsprechend einer vorgeschriebenen Regel. Typische Fragen sind: - Wie ändert sich die Wahrscheinlichkeit, dass ein von einem zufälligen Graphen-Prozess erzeugter Graph zusammenhängend ist? - Wann erfolgt der Phasenübergang? - Wie groß ist die größte Komponente? In dieser Habilitationsschrift werden diese Fragen über zufällige Graphen-Prozesse mit Gradbeschränkungen beantwortet. Dafür werden probabilistische Methoden, insbesondere Differentialgleichungsmethode, Verzweigungsprozesse, Singularitätsanalyse und Fourier-Transformationen, angewendet. / This thesis focuses on two kinds of discrete structures: planar structures, such as planar graphs and subclasses of them, and random graphs, particularly graphs generated by random processes. We study first planar structures from the following aspects. - How many of them are there (exactly or asymptotically)? - How can we efficiently sample a random instance uniformly at random? - What properties does a random planar structure have, with high probability? To answer these questions we decompose the planar structures along the connectivity. For the asymptotic enumeration we interpret the decomposition in terms of generating functions and derive the asymptotic number, using singularity analysis. For the exact enumeration and the uniform generation we use the recursive method. For typical properties of random planar structures we use the probabilistic method, together with the asymptotic numbers. Next we study random graph processes. Random graphs were first introduced by Erdos and Renyi and studied extensively since. A random graph process is a Markov chain whose stages are graphs on a given vertex set. It starts with an empty graph, and in each step a new graph is obtained from a current graph by adding a new edge according to a prescribed rule. Recently random graph processes with degree restrictions received much attention. In the thesis, we study random graph processes where the minimum degree grows quite quickly with the following questions in mind: - How does the connectedness of a graph generated by a random graph process change as the number of edges increases? - When does the phase transition occur? - How big is the largest component? To investigate the random graph processes we use the probabilistic method, Wormald''s differential equation method, multi-type branching processes, and the singularity analysis.

Zufallsgraphen sind Graphen, die durch einen zufälligen Prozess erzeugt werden. Ein im Zusammenhang mit Zufallsgraphen häufig auftretendes Phänomen ist, dass sich die typischen Eigenschaften eines Graphen durch Hinzufügen einer relativ kleinen Anzahl von zufälligen Kanten radikal verändern. Wir betrachten den Zufallsgraphen G(n,p), der n Knoten enthält und in dem zwei Knoten unabhängig und mit Wahrscheinlichkeit p durch eine Kante verbunden sind. Erdös und Rényi zeigten, dass ein Graph für p = c/n und c < 1 mit hoher Wahrscheinlichkeit aus Komponenten mit O(log n) Knoten besteht. Für p = c/n und c > 1 enthält G(n,p) mit hoher Wahrscheinlichkeit genau eine Komponente mit Theta(n) Knoten, welche viel größer als alle anderen Komponenten ist. Dieser Punkt in der Entwicklung des Graphen, an dem sich die Komponentenstruktur durch eine kleine Erhöhung der Anzahl von Kanten stark verändert, wird Phasenübergang genannt. Wenn p = (1+epsilon)/n, wobei epsilon eine Funktion von n ist, die gegen 0 geht, sind wir in der kritischen Phase, welche eine der interessantesten Phasen der Entwicklung des Zufallsgraphen ist. In dieser Arbeit betrachten wir drei verschiedene Modelle von Zufallsgraphen. In Kapitel 4 studieren wir den Minimalgrad-Graphenprozess. In diesem Prozess werden sukzessive Kanten vw hinzugefügt, wobei v ein zuällig ausgewählter Knoten von minimalem Grad ist. Wir beweisen, dass es in diesem Graphenprozess einen Phasenübergang, und wie im G(n,p) einen Doppelsprung, gibt. Die zwei anderen Modelle sind Zufallsgraphen mit einer vorgeschriebenen Gradfolge und zufällige gerichtete Graphen. Für diese Modelle wurde bereits in den Arbeiten von Molloy und Reed (1995), Karp (1990) und Luczak (1990) gezeigt, dass es einen Phasenübergang bezüglich der Komponentenstruktur gibt. In dieser Arbeit untersuchen wir in Kapitel 5 und 6 die kritische Phase dieser Prozesse genauer, und zeigen, dass sich diese Modelle ähnlich zum G(n,p) verhalten. / Random graphs are graphs which are created by a random process. A common phenomenon in random graphs is that the typical properties of a graph change radically by the addition of a relatively small number of random edges. This phenomenon was first investigated in the seminal papers of Erdös and Rényi. We consider the graph G(n,p) which contains n vertices, and where any two vertices are connected by an edge independently with probability p. Erdös and Rényi showed that if p = c/n$ and c < 1, then with high probability G(n,p) consists of components with O(log n) vertices. If p = c/n$ and c>1, then with high probability G(n,p) contains exactly one component, called the giant component, with Theta(n) vertices, which is much larger than all other components. The point at which the giant component is formed is called the phase transition. If we let $p = (1+epsilon)/n$, where epsilon is a function of n tending to 0, we are in the critical phase of the random graph, which is one of the most interesting phases in the evolution of the random graph. In this case the structure depends on how fast epsilon tends to 0. In this dissertation we consider three different random graph models. In Chapter 4 we consider the so-called minimum degree graph process. In this process edges vw are added successively, where v is a randomly chosen vertex with minimum degree. We prove that a phase transition occurs in this graph process as well, and also that it undergoes a double jump, similar to G(n,p). The two other models we will consider, are random graphs with a given degree sequence and random directed graphs. In these models the point of the phase transition has already been found, by Molloy and Reed (1995), Karp (1990) and Luczak (1990). In Chapter 5 and 6 we investigate the critical phase of these processes, and show that their behaviour resembles G(n,p).

We consider a generalisation of proper vertex colouring of graphs, referred to as improper colouring, in which each vertex can only be adjacent to a bounded number t of vertices with the same colour, and we study this type of graph colouring problem in several different settings. The thesis is divided into six chapters. In Chapter 1, we outline previous work in the area of improper colouring. In Chapters 2 and 3, we consider improper colouring of unit disk graphs -- a topic motivated by applications in telecommunications -- and take two approaches, first an algorithmic one and then an average-case analysis. In Chapter 4, we study the asymptotic behaviour of the improper chromatic number for the classical Erdos-Renyi model of random graphs. In Chapter 5, we discuss acyclic improper colourings, a specialisation of improper colouring, for graphs of bounded maximum degree. Finally, in Chapter 6, we consider another type of colouring, frugal colouring, in which no colour appears more than a bounded number of times in any neighbourhood. Throughout the thesis, we will observe a gradient of behaviours: for random unit disk graphs and "large" unit disk graphs, we can greatly reduce the required number of colours relative to proper colouring; in Erdos-Renyi random graphs, we do gain some improvement but only when t is relatively large; for acyclic improper chromatic numbers of bounded degree graphs, we discern an asymptotic difference in only a very narrow range of choices for t.

Η παρούσα διπλωματική εργασία βρίσκεται στον ευρύτερο χώρο της μαθηματικής στατιστικής θεωρίας των γραφημάτων. Κύριος στόχος μας, όπως αναφέρει και ο τίτλος, είναι η μοντελοποίηση γραφημάτων, με απώτερο σκοπό την προσαρμογή, προσομοίωση και διάγνωση αυτών μέσω μοντέλων εκθετικών τυχαίων γραφημάτων. Το πρώτο κεφάλαιο δίνει μια συνοπτική παρουσίαση της διατύπωσης του προβλήματος και της θεωρίας των μοντέλων των εκθετικών τυχαίων γραφημάτων. Η βασική ιδέα είναι να θεωρήσουμε ως τυχαίες μεταβλητές τους δυνατούς δεσμούς μεταξύ των κόμβων ενός δοθέντος γραφήματος. Η γενική μορφή ενός μοντέλου εκθετικά τυχαίου γραφήματος καθορίζεται από κάποιες υποθέσεις σχετικές με τις εξαρτήσεις μεταξύ αυτών των τυχαίων μεταβλητών. Παρουσιάζουμε κάποιες διαφορετικές υποθέσεις εξάρτησης και τα αντίστοιχα μοντέλα, όπως τα γραφημάτα Bernoulli, τα δυαδικώς - ανεξάρτητα και τα τυχαία γραφήματα Markov. Επίσης, εξετάζουμε την ενσωμάτωση των χαρακτηριστικών, που μπορούν να έχουν οι κόμβοι, σε μοντέλα κοινωνικής επιλογής, δηλαδή, σε περιπτώσεις που οι συνδέσεις του γραφήματος μπορούν να προβλέψουν τα χαρακτηριστικά των κόμβων. Συνοψίζουμε κάποιες καινούργιες υποθέσεις εξάρτησης, που είναι πολυπλοκότερες των πρώτων τέτοιων υποθέσεων της σχετικής βιβλιογραφίας. Συζητούμε τις διαδικασίες της στατιστικής εκτίμησης, συμπεριλαμβανομένων των νέων μεθόδων για την εκτίμηση της μέγιστης πιθανοφάνειας Monte Carlo. Τέλος, παρουσιάζουμε τις νέες προδιαγραφές για μοντέλα εκθετικών τυχαίων γραφημάτων, που έχουν προτείνει οι Snijders et al., οι οποίες βελτιώνουν σημαντικά τα αποτελέσματα της προσαρμογής εμπειρικών δεδομένων για εκθετικά μοντέλα ομοιογενών τυχαίων γραφημάτων Markov. Επιπλέον, οι νέες αυτές προδιαγραφές μας βοηθούν να αποφύγουμε το πρόβλημα του σχεδόν-εκφυλισμού, που συχνά παρεμβάλλεται στη διαδικασία της προσαρμογής μοντέλων εκθετικών τυχαίων γραφημάτων Markov, ιδιαίτερα όταν αυτά προέρχονται από εμπειρικά δεδομένα, που έχουν υψηλό βαθμό μεταβατικότητας. Η μελέτη μιας τέτοιας νέας στατιστικής με υψηλότερης τάξης μεταβατικότητα επιτρέπει την εκτίμηση των παραμέτρων των μοντέλων των εκθετικών γραφημάτων σε πολλές (αλλά όχι όλες) περιπτώσεις, στις οποίες διαφορετικά θα ήταν αδύνατο να εκτιμηθούν οι παράμετροι των μοντέλων των ομοιογενών γραφημάτων Markov. Στο δεύτερο, τρίτο και τέταρτο κεφάλαιο της εργασίας εφαρμόζουμε τις παραπάνω μεθόδους, αντιστοίχως, για τρείς αναλύσεις εμπειρικών δεδομένων: το δίκτυο Florentine, το δίκτυο Faux Magnolia High και τα δίκτυα IPRED και SWPAT. Σε αυτά τα κεφάλαια, παρουσιάζουμε τις διαδικασίες της προσαρμογής, προσομοίωσης και διάγνωσης με παράθεση των αντίστοιχων εντολών, χρησιμοποιώντας τα πακέτα statnet - ermg και sna, τα οποία δουλεύουν στο περιβάλλον του πακέτου ελεύθερου λογισμικού R. Τέλος, στο παράρτημα της εργασίας δίνουμε μια σύντομη εισαγωγή στο περιβάλλον R και σε κάποιες γενικές εντολές αυτού. / This specific project has to do with mathematical statistical graph theory. Our main target is to fit, simulate and diagnose models through exponential random graph models. In the first chapter we give a short presentation of the problem and the theory of exponential random graph models. The main idea is to consider each tie of a given network (graph) as a random variable. The general form of an exponential random graph model is defined from some relative assumptions that have to do with the dependence between those random variables. We present some different dependence assumptions and the corresponding models, such as Bernoulli graphs, dyadic-independent and Markov random graphs. We also examine the incorporation of the characteristics that a node may have in social networks. We also discuss the process of statistical estimation, including three new methods for the estimation of Monte Carlo maximum likelihood. Finally, we present new specifications for exponential random graph models, which Snijders et al. have proposed. These new specifications allow us to avoid the problem of degeneration. In the second, third and fourth chapter we apply the above methods in order to analyze Florentine network data, Faux Magnolia High data and IPred And Swpat data. In those chapters, we present the procedures of fit, simulate and diagnose exponential random graph models displaying the corresponding commands of statnet-ergm and sna packages that work in R. Finally we give a short introduction to R and to some relative commands.