Computationally efficient inference for latent position network models

Latent position models are widely used for the analysis of networks in a variety of research fields. In fact, these models possess a number of desirable theoretical properties, and are particularly easy to interpret. However, statistical methodologies to fit these models generally incur a computational cost which grows with the square of the number of nodes in the graph. This makes the analysis of large social networks impractical. In this paper, we propose a new method characterised by a linear computational complexity, which can be used to fit latent position models on networks of several tens of thousands nodes. Our approach relies on an approximation of the likelihood function, where the amount of noise introduced by the approximation can be arbitrarily reduced at the expense of computational efficiency. We establish several theoretical results that show how the likelihood error propagates to the invariant distribution of the Markov chain Monte Carlo sampler. In particular, we demonstrate that one can achieve a substantial reduction in computing time and still obtain a good estimate of the latent structure. Finally, we propose applications of our method to simulated networks and to a large coauthorships network, highlighting the usefulness of our approach.Comment: 39 pages, 10 figures, 1 tabl

Continuous Latent Position Models for Instantaneous Interactions

We create a framework to analyse the timing and frequency of instantaneous interactions between pairs of entities. This type of interaction data is especially common nowadays, and easily available. Examples of instantaneous interactions include email networks, phone call networks and some common types of technological and transportation networks. Our framework relies on a novel extension of the latent position network model: we assume that the entities are embedded in a latent Euclidean space, and that they move along individual trajectories which are continuous over time. These trajectories are used to characterize the timing and frequency of the pairwise interactions. We discuss an inferential framework where we estimate the individual trajectories from the observed interaction data, and propose applications on artificial and real data.Comment: 33 page

A mixture of experts model for rank data with applications in election studies

A voting bloc is defined to be a group of voters who have similar voting preferences. The cleavage of the Irish electorate into voting blocs is of interest. Irish elections employ a ``single transferable vote'' electoral system; under this system voters rank some or all of the electoral candidates in order of preference. These rank votes provide a rich source of preference information from which inferences about the composition of the electorate may be drawn. Additionally, the influence of social factors or covariates on the electorate composition is of interest. A mixture of experts model is a mixture model in which the model parameters are functions of covariates. A mixture of experts model for rank data is developed to provide a model-based method to cluster Irish voters into voting blocs, to examine the influence of social factors on this clustering and to examine the characteristic preferences of the voting blocs. The Benter model for rank data is employed as the family of component densities within the mixture of experts model; generalized linear model theory is employed to model the influence of covariates on the mixing proportions. Model fitting is achieved via a hybrid of the EM and MM algorithms. An example of the methodology is illustrated by examining an Irish presidential election. The existence of voting blocs in the electorate is established and it is determined that age and government satisfaction levels are important factors in influencing voting in this election.Comment: Published in at http://dx.doi.org/10.1214/08-AOAS178 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Active Ranking using Pairwise Comparisons

This paper examines the problem of ranking a collection of objects using pairwise comparisons (rankings of two objects). In general, the ranking of $n$ objects can be identified by standard sorting methods using $n log_2 n$ pairwise comparisons. We are interested in natural situations in which relationships among the objects may allow for ranking using far fewer pairwise comparisons. Specifically, we assume that the objects can be embedded into a $d$-dimensional Euclidean space and that the rankings reflect their relative distances from a common reference point in $R^d$. We show that under this assumption the number of possible rankings grows like $n^{2d}$ and demonstrate an algorithm that can identify a randomly selected ranking using just slightly more than $d log n$ adaptively selected pairwise comparisons, on average. If instead the comparisons are chosen at random, then almost all pairwise comparisons must be made in order to identify any ranking. In addition, we propose a robust, error-tolerant algorithm that only requires that the pairwise comparisons are probably correct. Experimental studies with synthetic and real datasets support the conclusions of our theoretical analysis.Comment: 17 pages, an extended version of our NIPS 2011 paper. The new version revises the argument of the robust section and slightly modifies the result there to give it more impac

Model-based clustering for multivariate partial ranking data

International audienceThis paper proposes the first model-based clustering algorithm dedicated to multivariate partial ranking data. This is an extension of the Insertion Sorting Rank (isr) model for ranking data, which is a meaningful and effective model obtained by modelling the ranking generating process assumed to be a sorting algorithm. The heterogeneity of the rank population is modelled by a mixture of isr, whereas conditional independence assumption allows the extension to multivariate ranking. Maximum likelihood estimation is performed through a SEM-Gibbs algorithm, and partial rankings are considered as missing data, what allows to simulate them during the estimation process. After having validated the estimation algorithm on simulations, three real datasets are studied: the 1980 American Psychological Association (APA) presidential election votes, the results of French students to a general knowledge test and the votes of the European countries to the Eurovision song contest. For each application, the proposed model shows relevant adequacy and leads to significant interpretation. In particular, regional alliances between European countries are exhibited in the Eurovision contest, which are often suspected but never proved.Nous proposons le premier modèle de classification automatique pour données de rang multivariées potentiellement incomplètes. Ce modèle est une extension du modèle Insertion Sorting Rank (isr) pour données de rang, qui est un modèle efficace et signifiant obtenu en modélisant le processus de génération des données. L'hétérogénéité des données est traitée à l'aide d'un modèle de mélange, tandis qu'une hypothèse classique d'indépendance conditionnelle permet de prendre en compte les rangs multivariés. L'estimation des paramètres du modèle est réalisée par maximum de vraisemblance à l'aide d'un algorithme SEM-Gibbs. Les données incomplètes sont considérées comme des données manquantes, ce qui permet de les simuler durant le processus d'estimation. Après avoir validé la stratégie d'estimation sur données simulées, trois jeux de données ont été étudiés : les votes lors de l'élection du président de l'American Psychological Association de 1980, les résultats d'étudiants français lors d'un test de culture générale, et les votes des pays lors du concours de l'Eurovision. Pour chaque application, le modèle proposé a montré une très bonne qualité d'ajustement et à conduit à des interprétations intéressantes. Notamment, pour le concours de l'Eurovision, nous avons mis à jour des alliances géographiques entre pays voisins, ce qui a souvent été suspecté pour ce concours mais jamais prouvé