Hawkes processes are often applied to model dependence and interaction phenomena in multivariate event data sets, such as neuronal spike trains, social interactions, and financial transactions. In the nonparametric setting, learning the temporal dependence structure of Hawkes processes is generally a computationally expensive task, all the more with Bayesian estimation methods. In particular, for multivariate nonlinear Hawkes processes, Monte-Carlo Markov Chain (MCMC) methods used to sample from the posterior distribution do not scale well to the dimension of the process. Recently, efficient algorithms targeting a mean-field variational approximation of the posterior distribution have been proposed, however, these methods do not allow to perform model selection on the graph of interactions of the Hawkes model. In this work, we propose a novel adaptive Bayesian variational method that performs model selection and can estimate a sparse graphical parameter. For the popular sigmoid Hawkes processes, we design a parallel algorithm which is scalable to high-dimensional point processes and large sequences of events. Furthermore, we unify existing variational Bayes approaches under a general nonparametric inference framework, and analyse the asymptotic properties of these methods under easily verifiable conditions on the prior, the variational class, and the nonlinear model. Finally, through an extensive set of numerical simulations, we demonstrate that our method is able to adapt to the dimensionality of the parameter of the Hawkes process, and is partially robust to certain types of model misspecification.

Abstract Dynamic networks are ubiquitous for modelling sequential graph-structured data, e.g., brain connectivity, population migrations, and social networks. In this work, we consider the discrete-time framework of dynamic networks and aim at detecting change-points, i.e., abrupt changes in the structure or attributes of the graph snapshots. This task is often termed network change-point detection and has numerous applications, such as market phase discovery, fraud detection, and activity monitoring. In this work, we propose a data-driven method that can adapt to the specific network domain, and be used to detect distribution changes with no delay and in an online setting. Our algorithm is based on a siamese graph neural network, designed to learn a graph similarity function on the graph snapshots from the temporal network sequence. Without any prior knowledge on the network generative distribution and the type of change-points, our learnt similarity function allows to more effectively compare the current graph and its recent history, compared to standard graph distances or kernels. Moreover, our method can be applied to a large variety of network data, e.g., networks with edge weights or node attributes. We test our method on synthetic and real-world dynamic network data, and demonstrate that it is able to perform online network change-point detection in diverse settings. Besides, we show that it requires a shorter data history to detect changes than most existing state-of-the-art baselines.

We study the problem of k-way clustering in signed graphs. Considerable attention in recent years has been devoted to analyzing and modeling signed graphs, where the affinity measure between nodes takes either positive or negative values. Recently, Cucuringu et al. (2019) proposed a spectral method, namely SPONGE (Signed Positive over Negative Generalized Eigenproblem), which casts the clustering task as a generalized eigenvalue problem optimizing a suitably defined objective function. This approach is motivated by social balance theory, where the clustering task aims to decompose a given network into disjoint groups, such that individuals within the same group are connected by as many positive edges as possible, while individuals from different groups are mainly connected by negative edges. Through extensive numerical experiments, SPONGE was shown to achieve state-of-the-art empirical performance. On the theoretical front, Cucuringu et al. (2019) analyzed SPONGE, as well as the popular Signed Laplacian based spectral method under the setting of a Signed Stochastic Block Model, for k=2 equal-sized clusters, in the regime where the graph is moderately dense. In this work, we build on the results in Cucuringu et al. (2019) on two fronts for the normalized versions of SPONGE and the Signed Laplacian. Firstly, for both algorithms, we extend the theoretical analysis in Cucuringu et al. (2019) to the general setting of k >= 2 unequal-sized clusters in the moderately dense regime. Secondly, we introduce regularized versions of both methods to handle sparse graphs -- a regime where standard spectral methods are known to underperform -- and provide theoretical guarantees under the same setting of a Signed Stochastic Block Model. To the best of our knowledge, regularized spectral methods have so far not been considered in the setting of clustering signed graphs. We complement our theoretical results with an extensive set of numerical experiments on synthetic data, and three real world data sets standard in the signed networks literature.