Multilayer Networks

In most natural and engineered systems, a set of entities interact with each other in complicated patterns that can encompass multiple types of relationships, change in time, and include other types of complications. Such systems include multiple subsystems and layers of connectivity, and it is important to take such "multilayer" features into account to try to improve our understanding of complex systems. Consequently, it is necessary to generalize "traditional" network theory by developing (and validating) a framework and associated tools to study multilayer systems in a comprehensive fashion. The origins of such efforts date back several decades and arose in multiple disciplines, and now the study of multilayer networks has become one of the most important directions in network science. In this paper, we discuss the history of multilayer networks (and related concepts) and review the exploding body of work on such networks. To unify the disparate terminology in the large body of recent work, we discuss a general framework for multilayer networks, construct a dictionary of terminology to relate the numerous existing concepts to each other, and provide a thorough discussion that compares, contrasts, and translates between related notions such as multilayer networks, multiplex networks, interdependent networks, networks of networks, and many others. We also survey and discuss existing data sets that can be represented as multilayer networks. We review attempts to generalize single-layer-network diagnostics to multilayer networks. We also discuss the rapidly expanding research on multilayer-network models and notions like community structure, connected components, tensor decompositions, and various types of dynamical processes on multilayer networks. We conclude with a summary and an outlook.Comment: Working paper; 59 pages, 8 figure

Dissémination de l’information et dynamique des opinions dans les réseaux sociaux

Our aim in this Ph. D. thesis is to study the diffusion of information as well as the opinion dynamics of users in social networks. Information diffusion models explore the paths taken by information being transmitted through a social network in order to understand and analyze the relationships between users in such network, leading to a better comprehension of human relations and dynamics. This thesis is based on both sides of information diffusion: first by developing mathematical theories and models to study the relationships between people and information, and in a second time by creating tools to better exploit the hidden patterns in these relationships. The theoretical tools developed in this thesis are opinion dynamics models and information diffusion models, where we study the information flow from users in social networks, and the practical tools developed in this thesis are a novel community detection algorithm and a novel trend detection algorithm. We start by introducing an opinion dynamics model in which agents interact with each other about several distinct opinions/contents. In our framework, agents do not exchange all their opinions with each other, they communicate about randomly chosen opinions at each time. We show, using stochastic approximation algorithms, that under mild assumptions this opinion dynamics algorithm converges as time increases, whose behavior is ruled by how users choose the opinions to broadcast at each time. We develop next a community detection algorithm which is a direct application of this opinion dynamics model: when agents broadcast the content they appreciate the most. Communities are thus formed, where they are defined as groups of users that appreciate mostly the same content. This algorithm, which is distributed by nature, has the remarkable property that the discovered communities can be studied from a solid mathematical standpoint. In addition to the theoretical advantage over heuristic community detection methods, the presented algorithm is able to accommodate weighted networks, parametric and nonparametric versions, with the discovery of overlapping communities a byproduct with no mathematical overhead. In a second part, we define a general framework to model information diffusion in social networks. The proposed framework takes into consideration not only the hidden interactions between users, but as well the interactions between contents and multiple social networks. It also accommodates dynamic networks and various temporal effects of the diffusion. This framework can be combined with topic modeling, for which several estimation techniques are derived, which are based on nonnegative tensor factorization techniques. Together with a dimensionality reduction argument, this techniques discover, in addition, the latent community structure of the users in the social networks. At last, we use one instance of the previous framework to develop a trend detection algorithm designed to find trendy topics in a social network. We take into consideration the interaction between users and topics, we formally define trendiness and derive trend indices for each topic being disseminated in the social network. These indices take into consideration the distance between the real broadcast intensity and the maximum expected broadcast intensity and the social network topology. The proposed trend detection algorithm uses stochastic control techniques in order calculate the trend indices, is fast and aggregates all the information of the broadcasts into a simple one-dimensional process, thus reducing its complexity and the quantity of necessary data to the detection. To the best of our knowledge, this is the first trend detection algorithm that is based solely on the individual performances of topicsLa dissémination d'information explore les chemins pris par l'information qui est transmise dans un réseau social, afin de comprendre et modéliser les relations entre les utilisateurs de ce réseau, ce qui permet une meilleur compréhension des relations humaines et leurs dynamique. Même si la priorité de ce travail soit théorique, en envisageant des aspects psychologiques et sociologiques des réseaux sociaux, les modèles de dissémination d'information sont aussi à la base de plusieurs applications concrètes, comme la maximisation d'influence, la prédication de liens, la découverte des noeuds influents, la détection des communautés, la détection des tendances, etc. Cette thèse est donc basée sur ces deux facettes de la dissémination d'information: nous développons d'abord des cadres théoriques mathématiquement solides pour étudier les relations entre les personnes et l'information, et dans un deuxième moment nous créons des outils responsables pour une exploration plus cohérente des liens cachés dans ces relations. Les outils théoriques développés ici sont les modèles de dynamique d'opinions et de dissémination d'information, où nous étudions le flot d'informations des utilisateurs dans les réseaux sociaux, et les outils pratiques développés ici sont un nouveau algorithme de détection de communautés et un nouveau algorithme de détection de tendances dans les réseaux sociau