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

National audienceIn 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.Le modèle à blocs stochastiques (SBM) décrit les interactions entre les sommets d'un graphe selon une approche probabiliste, basée sur des classes latentes. SBM fait l'hypothèse implicite que le graphe est stationnaire. Par conséquent, les interactions entre deux classes sont supposées avoir la même intensité pendant toute la période d'activité. Pour relaxer l'hypothèse de stationnarité, nous proposons une partition de l'horizon temporel en sous intervalles disjoints, chacun de même longueur. Ensuite, nous proposons une extension de SBM qui nous permet de classer en même temps les sommets du graphe et les intervalles de temps où les interactions ont lieu. Le nombre de classes latentes (K pour les sommets, D pour les intervalles de temps) est enfin obtenu à travers la maximisation de la vraisemblance intégrée des données complétées (ICL exacte). Après avoir testé le modèle sur des données simulées, nous traitons un cas réel. Pendant une journée, les interactions parmi les participants de la conférence HCM Hypertext (Turin, 29 Juin – 1er Juillet 2009) ont été traitées. Notre méthodologie nous a permis d'obtenir une classifications intéressante des 24 heures: les moments de rencontre tels que les pauses café ou buffets ont bien été détectés. La complexité de l'algorithme de recherche, linéaire en fonction du nombre initial de clusters ($K_{max}$ et $D_{max}$ respectivement), nous oriente vers l'utilisation d'instruments avancés de classification, pour réduire le nombre attendu de classes latentes et ainsi pouvoir utiliser le modèle pour des réseaux de grand dimension

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

The stochastic block model (SBM) is a flexible probabilistic tool that can be used to model interactions between clusters of nodes in a network. However, it does not account for interactions of time varying intensity between clusters. The extension of the SBM developed in this paper addresses this shortcoming through a temporal partition: assuming interactions between nodes are recorded on fixed-length time intervals, the inference procedure associated with the model we propose allows to cluster simultaneously the nodes of the network and the time intervals. The number of clusters of nodes and of time intervals, as well as the memberships to clusters, are obtained by maximizing an exact integrated complete-data likelihood, relying on a greedy search approach. Experiments on simulated and real data are carried out in order to assess the proposed methodology

Block modelling in dynamic networks with non-homogeneous Poisson processes and exact ICL

We develop a model in which interactions between nodes of a dynamic network are counted by non homogeneous Poisson processes. In a block modelling perspective, nodes belong to hidden clusters (whose number is unknown) and the intensity functions of the counting processes only depend on the clusters of nodes. In order to make inference tractable we move to discrete time by partitioning the entire time horizon in which interactions are observed in fixed-length time sub-intervals. First, we derive an exact integrated classification likelihood criterion and maximize it relying on a greedy search approach. This allows to estimate the memberships to clusters and the number of clusters simultaneously. Then a maximum-likelihood estimator is developed to estimate non parametrically the integrated intensities. We discuss the over-fitting problems of the model and propose a regularized version solving these issues. Experiments on real and simulated data are carried out in order to assess the proposed methodology

Statistical clustering of temporal networks through a dynamic stochastic block model

Statistical node clustering in discrete time dynamic networks is an emerging field that raises many challenges. Here, we explore statistical properties and frequentist inference in a model that combines a stochastic block model (SBM) for its static part with independent Markov chains for the evolution of the nodes groups through time. We model binary data as well as weighted dynamic random graphs (with discrete or continuous edges values). Our approach, motivated by the importance of controlling for label switching issues across the different time steps, focuses on detecting groups characterized by a stable within group connectivity behavior. We study identifiability of the model parameters, propose an inference procedure based on a variational expectation maximization algorithm as well as a model selection criterion to select for the number of groups. We carefully discuss our initialization strategy which plays an important role in the method and compare our procedure with existing ones on synthetic datasets. We also illustrate our approach on dynamic contact networks, one of encounters among high school students and two others on animal interactions. An implementation of the method is available as a R package called dynsbm

The Block Point Process Model for Continuous-Time Event-Based Dynamic Networks

We consider the problem of analyzing timestamped relational events between a set of entities, such as messages between users of an on-line social network. Such data are often analyzed using static or discrete-time network models, which discard a significant amount of information by aggregating events over time to form network snapshots. In this paper, we introduce a block point process model (BPPM) for continuous-time event-based dynamic networks. The BPPM is inspired by the well-known stochastic block model (SBM) for static networks. We show that networks generated by the BPPM follow an SBM in the limit of a growing number of nodes. We use this property to develop principled and efficient local search and variational inference procedures initialized by regularized spectral clustering. We fit BPPMs with exponential Hawkes processes to analyze several real network data sets, including a Facebook wall post network with over 3,500 nodes and 130,000 events.Comment: To appear at The Web Conference 201

Modeling sequences and temporal networks with dynamic community structures

In evolving complex systems such as air traffic and social organizations, collective effects emerge from their many components' dynamic interactions. While the dynamic interactions can be represented by temporal networks with nodes and links that change over time, they remain highly complex. It is therefore often necessary to use methods that extract the temporal networks' large-scale dynamic community structure. However, such methods are subject to overfitting or suffer from effects of arbitrary, a priori imposed timescales, which should instead be extracted from data. Here we simultaneously address both problems and develop a principled data-driven method that determines relevant timescales and identifies patterns of dynamics that take place on networks as well as shape the networks themselves. We base our method on an arbitrary-order Markov chain model with community structure, and develop a nonparametric Bayesian inference framework that identifies the simplest such model that can explain temporal interaction data.Comment: 15 Pages, 6 figures, 2 table

Bayesian nonparametrics for Sparse Dynamic Networks

We propose a Bayesian nonparametric prior for time-varying networks. To each node of the network is associated a positive parameter, modeling the sociability of that node. Sociabilities are assumed to evolve over time, and are modeled via a dynamic point process model. The model is able to (a) capture smooth evolution of the interaction between nodes, allowing edges to appear/disappear over time (b) capture long term evolution of the sociabilities of the nodes (c) and yield sparse graphs, where the number of edges grows subquadratically with the number of nodes. The evolution of the sociabilities is described by a tractable time-varying gamma process. We provide some theoretical insights into the model and apply it to three real world datasets.Comment: 10 pages, 8 figure

Optimization of Trading Physics Models of Markets

We describe an end-to-end real-time S&P futures trading system. Inner-shell stochastic nonlinear dynamic models are developed, and Canonical Momenta Indicators (CMI) are derived from a fitted Lagrangian used by outer-shell trading models dependent on these indicators. Recursive and adaptive optimization using Adaptive Simulated Annealing (ASA) is used for fitting parameters shared across these shells of dynamic and trading models