Variational Bayes model averaging for graphon functions and motif frequencies inference in W-graph models

W-graph refers to a general class of random graph models that can be seen as a random graph limit. It is characterized by both its graphon function and its motif frequencies. In this paper, relying on an existing variational Bayes algorithm for the stochastic block models along with the corresponding weights for model averaging, we derive an estimate of the graphon function as an average of stochastic block models with increasing number of blocks. In the same framework, we derive the variational posterior frequency of any motif. A simulation study and an illustration on a social network complete our work

Model Selection in Overlapping Stochastic Block Models

Networks are a commonly used mathematical model to describe the rich set of interactions between objects of interest. Many clustering methods have been developed in order to partition such structures, among which several rely on underlying probabilistic models, typically mixture models. The relevant hidden structure may however show overlapping groups in several applications. The Overlapping Stochastic Block Model (2011) has been developed to take this phenomenon into account. Nevertheless, the problem of the choice of the number of classes in the inference step is still open. To tackle this issue, we consider the proposed model in a Bayesian framework and develop a new criterion based on a non asymptotic approximation of the marginal log-likelihood. We describe how the criterion can be computed through a variational Bayes EM algorithm, and demonstrate its efficiency by running it on both simulated and real data.Comment: articl

Un modèle de sous-graphes aléatoires pour l'analyse de la dynamique d'un réseau historique

National audienceCes dernières années, de nombreux modèles de graphes aléatoires ont été proposé pour extraire des informations à partir de réseaux. Le principe consiste à chercher des groupes de sommets ayant des profils de connexions homogènes. La plupart de ces modèles sont adaptés pour les réseaux statiques, c'est-à-dire ne prenant pas en compte la dimension temporelle, mais pouvant gérer différents types d'arêtes, qu'elles soient binaires ou discrètes. Les modèles existant doivent donc être adaptés pour être capables de gérer l'aspect temporel. Nous nous intéressons ici au modèle de sous-graphes aléatoires proposé par [1]. Ce modèle permet de modéliser les sous-graphes connus d'un réseau à l'aide de structures latentes cachées. Nous proposons d'intégrer un processus à espace d'états à ce modèle afin d'autoriser les classes, ainsi que la topologie du réseau, à évaluer au cours du temps. L'inférence est réalisée par l'intermédiaire d'un algorithme Bayes variationnel. Un critère de sélection de modèles est également dérivé. Nous appliquons cette méthodologie à un réseau décrivant les relations entre évêques en Gaule Mérovingienne. Mots-clés. Réseau, sous-graphe, classification, algorithme VBEM. 1 Contexte de notre étude Ce travail a été motivé par la nécessité d'analyser un réseau historique évoluant dans le temps, pour lequel une partition des sommets est donnée et dont les arêtes sont de type catégorielle. L'analyse porte sur les relations interindividuelles tissées dans le cadre des conciles réunis en Gaule au VIe siècle. Le concile est une assemblée d'ecclésiastiques réunie sous l'autorité d'un évêque pour débattre des affaires ecclésiastiques. Pour le VIe siècle, 46 conciles furent réunis en Gaule, qui est alors, au gré des partages successoraux, divisée entre plusieurs royaumes : Neustrie, Austrasie et Bourgogne, la Bourgogne et l'Aquitaine changeant plusieurs fois de mains au cours du VIe siècle. La base de données regroupe 1331 personnages ayant occupé une ou plusieurs fonctions en Gaule entre 480 et 614 et les individus pour lesquels nous savons qu'ils ont eu des liens relationnels avec d'autres que ce soient des liens de parenté ou de connaissance. D'un point de vue historique, l'analyse et la comparaison de la structure organisationnelle de chacun des royaumes ou provinces pourrait fournir des éléments clés de compréhension de cette période

A Spectral Algorithm with Additive Clustering for the Recovery of Overlapping Communities in Networks

This paper presents a novel spectral algorithm with additive clustering designed to identify overlapping communities in networks. The algorithm is based on geometric properties of the spectrum of the expected adjacency matrix in a random graph model that we call stochastic blockmodel with overlap (SBMO). An adaptive version of the algorithm, that does not require the knowledge of the number of hidden communities, is proved to be consistent under the SBMO when the degrees in the graph are (slightly more than) logarithmic. The algorithm is shown to perform well on simulated data and on real-world graphs with known overlapping communities.Comment: Journal of Theoretical Computer Science (TCS), Elsevier, A Para\^itr

MAGMA : inference and prediction using multi-task Gaussian processes with common mean

A novel multi-task Gaussian process (GP) framework is proposed, by using a common mean process for sharing information across tasks. In particular, we investigate the problem of time series forecasting, with the objective to improve multiple-step-ahead predictions. The common mean process is defined as a GP for which the hyper-posterior distribution is tractable. Therefore an EM algorithm is derived for handling both hyper-parameters optimisation and hyper-posterior computation. Unlike previous approaches in the literature, the model fully accounts for uncertainty and can handle irregular grids of observations while maintaining explicit formulations, by modelling the mean process in a unified GP framework. Predictive analytical equations are provided, integrating information shared across tasks through a relevant prior mean. This approach greatly improves the predictive performances, even far from observations, and may reduce significantly the computational complexity compared to traditional multi-task GP models. Our overall algorithm is called MAGMA (standing for Multi tAsk GPs with common MeAn). The quality of the mean process estimation, predictive performances, and comparisons to alternatives are assessed in various simulated scenarios and on real datasets

Minimising pain in farm animals: the 3S approach - ‘Suppress, Substitute, Soothe'

Recently, the French National Institute for Agricultural Research appointed an expert committee to review the issue of pain in food-producing farm animals. To minimise pain, the authors developed a ‘3S' approach accounting for ‘Suppress, Substitute and Soothe' by analogy with the ‘3Rs' approach of ‘Reduction, Refinement and Replacement' applied in the context of animal experimentation. Thus, when addressing the matter of pain, the following steps and solutions could be assessed, in the light of their feasibility (technical constraints, logistics and regulations), acceptability (societal and financial aspects) and availability. The first solution is to suppress any source of pain that brings no obvious advantage to the animals or the producers, as well as sources of pain for which potential benefits are largely exceeded by the negative effects. For instance, tail docking of cattle has recently been eliminated. Genetic selection on the basis of resistance criteria (as e.g. for lameness in cattle and poultry) or reduction of undesirable traits (e.g. boar taint in pigs) may also reduce painful conditions or procedures. The second solution is to substitute a technique causing pain by another less-painful method. For example, if dehorning cattle is unavoidable, it is preferable to perform it at a very young age, cauterising the horn bud. Animal management and constraint systems should be designed to reduce the risk for injury and bruising. Lastly, in situations where pain is known to be present, because of animal management procedures such as dehorning or castration, or because of pathology, for example lameness, systemic or local pharmacological treatments should be used to soothe pain. These treatments should take into account the duration of pain, which, in the case of some management procedures or diseases, may persist for longer periods. The administration of pain medication may require the intervention of veterinarians, but exemptions exist where breeders are allowed to use local anaesthesia (e.g. castration and dehorning in Switzerland). Extension of such exemptions, national or European legislation on pain management, or the introduction of animal welfare codes by retailers into their meat products may help further developments. In addition, veterinarians and farmers should be given the necessary tools and information to take into account animal pain in their management decision