arXiv subjects

Pierre Latouche

Publications and source records attributed to Pierre Latouche.

30 records · Page 2Linked to original sources

arXiv2017

Choosing the number of groups in a latent stochastic block model for dynamic networks

Latent stochastic block models are flexible statistical models that are widely used in social network analysis. In recent years, efforts have been made to extend these models to temporal dynamic networks, whereby the connections between nodes are observed at a number of different times. In this paper we extend the original stochastic block model by using a Markovian property to describe the evolution of nodes' cluster memberships over time. We recast the problem of clustering the nodes of the network into a model-based context, and show that the integrated completed likelihood can be evaluated analytically for a number of likelihood models. Then, we propose a scalable greedy algorithm to maximise this quantity, thereby estimating both the optimal partition and the ideal number of groups in a single inferential framework. Finally we propose applications of our methodology to both real and artificial datasets.

stat.ME↗

arXiv2017

Goodness of fit of logistic models for random graphs

Logistic regression is a natural and simple tool to understand how covariates contribute to explain the topology of a binary network. Once the model fitted, the practitioner is interested in the goodness-of-fit of the regression in order to check if the covariates are sufficient to explain the whole topology of the network and, if they are not, to analyze the residual structure. To address this problem, we introduce a generic model that combines logistic regression with a network-oriented residual term. This residual term takes the form of the graphon function of a W-graph. Using a variational Bayes framework, we infer the residual graphon by averaging over a series of blockwise constant functions. This approach allows us to define a generic goodness-of-fit criterion, which corresponds to the posterior probability for the residual graphon to be constant. Experiments on toy data are carried out to assess the accuracy of the procedure. Several networks from social sciences and ecology are studied to illustrate the proposed methodology.

stat.ME↗

arXiv2016

Bayesian Variable Selection for Globally Sparse Probabilistic PCA

Sparse versions of principal component analysis (PCA) have imposed themselves as simple, yet powerful ways of selecting relevant features of high-dimensional data in an unsupervised manner. However, when several sparse principal components are computed, the interpretation of the selected variables is difficult since each axis has its own sparsity pattern and has to be interpreted separately. To overcome this drawback, we propose a Bayesian procedure called globally sparse probabilistic PCA (GSPPCA) that allows to obtain several sparse components with the same sparsity pattern. This allows the practitioner to identify the original variables which are relevant to describe the data. To this end, using Roweis' probabilistic interpretation of PCA and a Gaussian prior on the loading matrix, we provide the first exact computation of the marginal likelihood of a Bayesian PCA model. To avoid the drawbacks of discrete model selection, a simple relaxation of this framework is presented. It allows to find a path of models using a variational expectation-maximization algorithm. The exact marginal likelihood is then maximized over this path. This approach is illustrated on real and synthetic data sets. In particular, using unlabeled microarray data, GSPPCA infers much more relevant gene subsets than traditional sparse PCA algorithms.

stat.ML↗

arXiv2016

Is the corporate elite disintegrating? Interlock boards and the Mizruchi hypothesis

This paper proposes an approach for comparing interlocked board networks over time to test for statistically significant change. In addition to contributing to the conversation about whether the Mizruchi hypothesis (that a disintegration of power is occurring within the corporate elite) holds or not, we propose novel methods to handle a longitudinal investigation of a series of social networks where the nodes undergo a few modifications at each time point. Methodologically, our contribution is twofold: we extend a Bayesian model hereto applied to compare two time periods to a longer time period, and we define and employ the concept of a hull of a sequence of social networks, which makes it possible to circumvent the problem of changing nodes over time.

physics.soc-ph↗

arXiv2015

Modelling time evolving interactions in networks through a non stationary extension of stochastic block models

In this paper, we focus on the stochastic block model (SBM),a probabilistic tool describing interactions between nodes of a network using latent clusters. The SBM assumes that the networkhas a stationary structure, in which connections of time varying intensity are not taken into account. In other words, interactions between two groups are forced to have the same features during the whole observation time. To overcome this limitation,we propose a partition of the whole time horizon, in which interactions are observed, and develop a non stationary extension of the SBM,allowing to simultaneously cluster the nodes in a network along with fixed time intervals in which the interactions take place. The number of clusters (K for nodes, D for time intervals) as well as the class memberships are finallyobtained through maximizing the complete-data integrated likelihood by means of a greedy search approach. After showing that the model works properly with simulated data, we focus on a real data set. We thus consider the three days ACM Hypertext conference held in Turin,June 29th - July 1st 2009. Proximity interactions between attendees during the first day are modelled and an interestingclustering of the daily hours is finally obtained, with times of social gathering (e.g. coffee breaks) recovered by the approach. Applications to large networks are limited due to the computational complexity of the greedy search which is dominated bythe number $K\_{max}$ and $D\_{max}$ of clusters used in the initialization. Therefore,advanced clustering tools are considered to reduce the number of clusters expected in the data, making the greedy search applicable to large networks.

stat.ML↗

arXiv2015

Graphs in machine learning: an introduction

Graphs are commonly used to characterise interactions between objects of interest. Because they are based on a straightforward formalism, they are used in many scientific fields from computer science to historical sciences. In this paper, we give an introduction to some methods relying on graphs for learning. This includes both unsupervised and supervised methods. Unsupervised learning algorithms usually aim at visualising graphs in latent spaces and/or clustering the nodes. Both focus on extracting knowledge from graph topologies. While most existing techniques are only applicable to static graphs, where edges do not evolve through time, recent developments have shown that they could be extended to deal with evolving networks. In a supervised context, one generally aims at inferring labels or numerical values attached to nodes using both the graph and, when they are available, node characteristics. Balancing the two sources of information can be challenging, especially as they can disagree locally or globally. In both contexts, supervised and un-supervised, data can be relational (augmented with one or several global graphs) as described above, or graph valued. In this latter case, each object of interest is given as a full graph (possibly completed by other characteristics). In this context, natural tasks include graph clustering (as in producing clusters of graphs rather than clusters of nodes in a single graph), graph classification, etc. 1 Real networks One of the first practical studies on graphs can be dated back to the original work of Moreno [51] in the 30s. Since then, there has been a growing interest in graph analysis associated with strong developments in the modelling and the processing of these data. Graphs are now used in many scientific fields. In Biology [54, 2, 7], for instance, metabolic networks can describe pathways of biochemical reactions [41], while in social sciences networks are used to represent relation ties between actors [66, 56, 36, 34]. Other examples include powergrids [71] and the web [75]. Recently, networks have also been considered in other areas such as geography [22] and history [59, 39]. In machine learning, networks are seen as powerful tools to model problems in order to extract information from data and for prediction purposes. This is the object of this paper. For more complete surveys, we refer to [28, 62, 49, 45]. In this section, we introduce notations and highlight properties shared by most real networks. In Section 2, we then consider methods aiming at extracting information from a unique network. We will particularly focus on clustering methods where the goal is to find clusters of vertices. Finally, in Section 3, techniques that take a series of networks into account, where each network is

stat.ML↗

arXiv2015

Exact ICL maximization in a non-stationary time extension of the latent block model for dynamic networks

The latent block model (LBM) is a flexible probabilistic tool to describe interactions between node sets in bipartite networks, but it does not account for interactions of time varying intensity between nodes in unknown classes. In this paper we propose a non stationary temporal extension of the LBM that clusters simultaneously the two node sets of a bipartite network and constructs classes of time intervals on which interactions are stationary. The number of clusters as well as the membership to classes are obtained by maximizing the exact complete-data integrated likelihood relying on a greedy search approach. Experiments on simulated and real data are carried out in order to assess the proposed methodology.

stat.ML↗

arXiv2015

Inferring structure in bipartite networks using the latent block model and exact ICL

We consider the task of simultaneous clustering of the two node sets involved in a bipartite network. The approach we adopt is based on use of the exact integrated complete likelihood for the latent block model. Using this allows one to infer the number of clusters as well as cluster memberships using a greedy search. This gives a model-based clustering of the node sets. Experiments on simulated bipartite network data show that the greedy search approach is vastly more scalable than competing Markov chain Monte Carlo based methods. Application to a number of real observed bipartite networks demonstrate the algorithms discussed.

stat.CO↗

arXiv2014

The random subgraph model for the analysis of an ecclesiastical network in Merovingian Gaul

In the last two decades many random graph models have been proposed to extract knowledge from networks. Most of them look for communities or, more generally, clusters of vertices with homogeneous connection profiles. While the first models focused on networks with binary edges only, extensions now allow to deal with valued networks. Recently, new models were also introduced in order to characterize connection patterns in networks through mixed memberships. This work was motivated by the need of analyzing a historical network where a partition of the vertices is given and where edges are typed. A known partition is seen as a decomposition of a network into subgraphs that we propose to model using a stochastic model with unknown latent clusters. Each subgraph has its own mixing vector and sees its vertices associated to the clusters. The vertices then connect with a probability depending on the subgraphs only, while the types of edges are assumed to be sampled from the latent clusters. A variational Bayes expectation-maximization algorithm is proposed for inference as well as a model selection criterion for the estimation of the cluster number. Experiments are carried out on simulated data to assess the approach. The proposed methodology is then applied to an ecclesiastical network in Merovingian Gaul. An R code, called Rambo, implementing the inference algorithm is available from the authors upon request.

stat.ME↗

arXiv2013

Activity date estimation in timestamped interaction networks

We propose in this paper a new generative model for graphs that uses a latent space approach to explain timestamped interactions. The model is designed to provide global estimates of activity dates in historical networks where only the interaction dates between agents are known with reasonable precision. Experimental results show that the model provides better results than local averages in dense enough networks

math.ST↗

arXiv2011

Overlapping stochastic block models with application to the French political blogosphere

Complex systems in nature and in society are often represented as networks, describing the rich set of interactions between objects of interest. Many deterministic and probabilistic clustering methods have been developed to analyze such structures. Given a network, almost all of them partition the vertices into disjoint clusters, according to their connection profile. However, recent studies have shown that these techniques were too restrictive and that most of the existing networks contained overlapping clusters. To tackle this issue, we present in this paper the Overlapping Stochastic Block Model. Our approach allows the vertices to belong to multiple clusters, and, to some extent, generalizes the well-known Stochastic Block Model [Nowicki and Snijders (2001)]. We show that the model is generically identifiable within classes of equivalence and we propose an approximate inference procedure, based on global and local variational techniques. Using toy data sets as well as the French Political Blogosphere network and the transcriptional network of Saccharomyces cerevisiae, we compare our work with other approaches.

stat.ME↗

arXiv2010

Variational Bayesian Inference and Complexity Control for Stochastic Block Models

It is now widely accepted that knowledge can be acquired from networks by clustering their vertices according to connection profiles. Many methods have been proposed and in this paper we concentrate on the Stochastic Block Model (SBM). The clustering of vertices and the estimation of SBM model parameters have been subject to previous work and numerous inference strategies such as variational Expectation Maximization (EM) and classification EM have been proposed. However, SBM still suffers from a lack of criteria to estimate the number of components in the mixture. To our knowledge, only one model based criterion, ICL, has been derived for SBM in the literature. It relies on an asymptotic approximation of the Integrated Complete-data Likelihood and recent studies have shown that it tends to be too conservative in the case of small networks. To tackle this issue, we propose a new criterion that we call ILvb, based on a non asymptotic approximation of the marginal likelihood. We describe how the criterion can be computed through a variational Bayes EM algorithm.

stat.AP↗