This thesis is comprised of three parts. The first part describes a novel framework for computing importance measures on graph vertices. The concept of a D-spectrum is introduced, based on vertex ranks within certain chains of nested sub-graphs. We show that the D- spectrum integrates the degree distribution and coreness information of the graph as two particular such chains. We prove that these spectra are realized as fixed points of certain monotone and contractive SDSs we call t-systems. Finally, we give a vertex deletion algorithm that efficiently computes D-spectra, and we illustrate their correlation with stochastic SIR-processes on real world networks. The second part deals with the topology of the intersection nerve for a bi-secondary structure, and its singular homology. A bi-secondary structure R, is a combinatorial object that can be viewed as a collection of cycles (loops) of certain at most tetravalent planar graphs. Bi-secondary structures arise naturally in the study of RNA riboswitches - molecules that have an MFE binary structural degeneracy. We prove that this loop nerve complex has a euclidean 3-space embedding characterized solely by H2(R), its second homology group. We show that this group is the only non-trivial one in the sequence and furthermore it is free abelian. The third part further describes the features of the loop nerve. We identify certain disjoint objects in the structure of R which we call crossing components (CC). These are non-trivial connected components of a graph that captures a particular non-planar embedding of R. We show that each CC contributes a unique generator to H2(R) and thus the total number of these crossing components in fact equals the rank of the second homology group. / Doctor of Philosophy / This Thesis is divided into three parts. The first part describes a novel mathematical framework for decomposing a real world network into layers. A network is comprised of interconnected nodes and can model anything from transportation of goods to the way the internet is organized. Two key numbers describe the local and global features of a network: the number of neighbors, and the number of neighbors in a certain layer, a node has. Our work shows that there are other numbers in-between the two, that better characterize a node. We also give explicit means of computing them. Finally, we show that these numbers are connected to the way information spreads on the network, uncovering a relation between the network’s structure and dynamics on said network. The last two parts of the thesis have a common theme and study the same mathematical object. In the first part of the two, we provide a new model for the way riboswtiches organize themselves. Riboswitches, are RNA molecules within a cell, that can take two mutually opposite conformations, depending on what function they need to perform within said cell. They are important from an evolutionary standpoint and are actively studied within that context, usually being modeled as networks. Our model captures the shapes of the two possible conformations, and encodes it within a mathematical object called a topological space. Once this is done, we prove that certain numbers that are attached to all topological spaces carry specific values for riboswitches. Namely, we show that the shapes of the two possible conformations for a riboswich are always characterized by a single integer. In the last part of the Thesis we identify what exactly in the structure of riboswitches contributes to this number being large or small. We prove that the more tangled the two conformations are, the larger the number. We can thus conclude that this number is directly proportional to how complex the riboswitch is.

Les contacts face-à-face entre individus permettent de caractériser les réseaux sociaux et jouent un rôle prépondérant dans la compréhension des mécanismes de propagation des épidémies dans une population. De récentes avancées technologiques ont rendu possible l'acquisition de données précises sur les interactions humaines. Cette thèse présente, dans un premier temps, l'analyse de données de contacts collectées trois années de suite (2011, 2012 et 2013) dans un lycée français entre des étudiants de classes préparatoires. L'analyse a montré que la plupart des contacts se produisent entre étudiants de même classe et que les structures des contacts sont très similaires d'un jour sur l'autre. Dans un second temps, on compare différentes méthodes de collecte de données qui permettent d'obtenir des informations de nature différente (par exemple existence d'un contact face-à-face vs existence d'une amitié).L'utilisation de données rapportant les amitiés entre les étudiants ne permet pas d'obtenir une bonne estimation du réseau de contact (i.e., les amitiés ne correspondent pas forcément à des contacts face-à-face et vice versa) résultant en une sous-estimation du risque épidémique dans cette population.Dans la dernière partie, nous essayons de reproduire les biais provenant du réseau d'amitié en échantillonnant le réseau de contact. Ceci pourrait nous donner des indications sur comment compenser ces biais et comment utiliser des données incomplètes pour obtenir des prédictions fiables sur le risque épidémique. / Face-to-face contacts between individuals contribute to shape social networks and play an important role in determining how infectious diseases can spread within a population. Recently, technological advances have made it possible to obtain accurate data on human interactions.This thesis first presents the analysis of contact data collected three years in a row (2011, 2012 and 2013) in a French high school among students of "classes préparatoires" (i.e., studies taking place after high school and preparing for admission to higher education colleges). The analysis showed that most contacts occur within students of same classes and that contact patterns are very similar from one day to the next.Then, we compare different methods of data collection which allow to gather information of different nature (for instance existence of a face-to-face contact vs existence of a friendship).The use of data reporting friendships does not allow to obtain a good estimation of the contact network (i.e., friendships do not correspond necessarily to face-to-face contacts and vice versa) resulting in an underestimation of the epidemic risk in that population.Finally, we try to reproduce the biases coming from the friendship network by sampling the contact network. This might give hints on how to compensate these biases and how to use the information contained in incomplete data sets to obtain accurate predictions of the epidemic risk.